<commit_before>"""
This module implements some special functions that commonly appear in
combinatorial contexts (e.g. in power series); in particular,
sequences of rational numbers such as Bernoulli and Fibonacci numbers.

Factorials, binomial coefficients and related functions are located in
the separate 'factorials' module.
"""

from sympy.core.function import Function, expand_mul
from sympy.core import S, Symbol, Rational, oo, Integer, C, Add, Dummy
from sympy.core.compatibility import as_int, SYMPY_INTS
from sympy.core.cache import cacheit
from sympy.functions.combinatorial.factorials import factorial

from sympy.mpmath import bernfrac
from sympy.mpmath.libmp import ifib as _ifib


def _product(a, b):
    p = 1
    for k in xrange(a, b + 1):
        p *= k
    return p

from sympy.utilities.memoization import recurrence_memo


# Dummy symbol used for computing polynomial sequences
_sym = Symbol('x')
_symbols = Function('x')


#----------------------------------------------------------------------------#
#                                                                            #
#                           Fibonacci numbers                                #
#                                                                            #
#----------------------------------------------------------------------------#

class fibonacci(Function):
    """
    Fibonacci numbers / Fibonacci polynomials

    The Fibonacci numbers are the integer sequence defined by the
    initial terms F_0 = 0, F_1 = 1 and the two-term recurrence
    relation F_n = F_{n-1} + F_{n-2}.

    The Fibonacci polynomials are defined by F_1(x) = 1,
    F_2(x) = x, and F_n(x) = x*F_{n-1}(x) + F_{n-2}(x) for n > 2.
    For all positive integers n, F_n(1) = F_n.

    * fibonacci(n) gives the nth Fibonacci number, F_n
    * fibonacci(n, x) gives the nth Fibonacci polynomial in x, F_n(x)

    Examples
    ========

    >>> from sympy import fibonacci, Symbol

    >>> [fibonacci(x) for x in range(11)]
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
    >>> fibonacci(5, Symbol('t'))
    t**4 + 3*t**2 + 1

    References
    ==========

    * http://en.wikipedia.org/wiki/Fibonacci_number
    * http://mathworld.wolfram.com/FibonacciNumber.html

    See Also
    ========

    lucas
    """

    @staticmethod
    def _fib(n):
        return _ifib(n)

    @staticmethod
    @recurrence_memo([None, S.One, _sym])
    def _fibpoly(n, prev):
        return (prev[-2] + _sym*prev[-1]).expand()

    @classmethod
    def eval(cls, n, sym=None):
        if n.is_Integer:
            n = int(n)
            if n < 0:
                return S.NegativeOne**(n + 1) * fibonacci(-n)
            if sym is None:
                return Integer(cls._fib(n))
            else:
                if n < 1:
                    raise ValueError("Fibonacci polynomials are defined "
                       "only for positive integer indices.")
                return cls._fibpoly(n).subs(_sym, sym)


class lucas(Function):
    """
    Lucas numbers

    Lucas numbers satisfy a recurrence relation similar to that of
    the Fibonacci sequence, in which each term is the sum of the
    preceding two. They are generated by choosing the initial
    values L_0 = 2 and L_1 = 1.

    * lucas(n) gives the nth Lucas number

    Examples
    ========

    >>> from sympy import lucas

    >>> [lucas(x) for x in range(11)]
    [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]

    References
    ==========

    * http://en.wikipedia.org/wiki/Lucas_number

    See Also
    ========

    fibonacci
    """

    @classmethod
    def eval(cls, n):
        if n.is_Integer:
            return fibonacci(n + 1) + fibonacci(n - 1)


#----------------------------------------------------------------------------#
#                                                                            #
#                           Bernoulli numbers                                #
#                                                                            #
#----------------------------------------------------------------------------#

class bernoulli(Function):
    r"""
    Bernoulli numbers / Bernoulli polynomials

    The Bernoulli numbers are a sequence of rational numbers
    defined by B_0 = 1 and the recursive relation (n > 0)::

                n
               ___
              \      / n + 1 \
          0 =  )     |       | * B .
              /___   \   k   /    k
              k = 0

    They are also commonly defined by their exponential generating
    function, which is x/(exp(x) - 1). For odd indices > 1, the
    Bernoulli numbers are zero.

    The Bernoulli polynomials satisfy the analogous formula::

                    n
                   ___
                  \      / n \         n-k
          B (x) =  )     |   | * B  * x   .
           n      /___   \ k /    k
                  k = 0

    Bernoulli numbers and Bernoulli polynomials are related as
    B_n(0) = B_n.

    We compute Bernoulli numbers using Ramanujan's formula::

                                   / n + 3 \
          B   =  (A(n) - S(n))  /  |       |
           n                       \   n   /

    where A(n) = (n+3)/3 when n = 0 or 2 (mod 6), A(n) = -(n+3)/6
    when n = 4 (mod 6), and::

                 [n/6]
                  ___
                 \      /  n + 3  \
          S(n) =  )     |         | * B
                 /___   \ n - 6*k /    n-6*k
                 k = 1

    This formula is similar to the sum given in the definition, but
    cuts 2/3 of the terms. For Bernoulli polynomials, we use the
    formula in the definition.

    * bernoulli(n) gives the nth Bernoulli number, B_n
    * bernoulli(n, x) gives the nth Bernoulli polynomial in x, B_n(x)

    Examples
    ========

    >>> from sympy import bernoulli

    >>> [bernoulli(n) for n in range(11)]
    [1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
    >>> bernoulli(1000001)
    0

    References
    ==========

    * http://en.wikipedia.org/wiki/Bernoulli_number
    * http://en.wikipedia.org/wiki/Bernoulli_polynomial

    See Also
    ========

    euler, bell
    """

    # Calculates B_n for positive even n
    @staticmethod
    def _calc_bernoulli(n):
        s = 0
        a = int(C.binomial(n + 3, n - 6))
        for j in xrange(1, n//6 + 1):
            s += a * bernoulli(n - 6*j)
            # Avoid computing each binomial coefficient from scratch
            a *= _product(n - 6 - 6*j + 1, n - 6*j)
            a //= _product(6*j + 4, 6*j + 9)
        if n % 6 == 4:
            s = -Rational(n + 3, 6) - s
        else:
            s = Rational(n + 3, 3) - s
        return s / C.binomial(n + 3, n)

    # We implement a specialized memoization scheme to handle each
    # case modulo 6 separately
    _cache = {0: S.One, 2: Rational(1, 6), 4: Rational(-1, 30)}
    _highest = {0: 0, 2: 2, 4: 4}

    @classmethod
    def eval(cls, n, sym=None):
        if n.is_Number:
            if n.is_Integer and n.is_nonnegative:
                if n is S.Zero:
                    return S.One
                elif n is S.One:
                    if sym is None:
                        return -S.Half
                    else:
                        return sym - S.Half
                # Bernoulli numbers
                elif sym is None:
                    if n.is_odd:
                        return S.Zero
                    n = int(n)
                    # Use mpmath for enormous Bernoulli numbers
                    if n > 500:
                        p, q = bernfrac(n)
                        return Rational(int(p), int(q))
                    case = n % 6
                    highest_cached = cls._highest[case]
                    if n <= highest_cached:
                        return cls._cache[n]
                    # To avoid excessive recursion when, say, bernoulli(1000) is
                    # requested, calculate and cache the entire sequence ... B_988,
                    # B_994, B_1000 in increasing order
                    for i in xrange(highest_cached + 6, n + 6, 6):
                        b = cls._calc_bernoulli(i)
                        cls._cache[i] = b
                        cls._highest[case] = i
                    return b
                # Bernoulli polynomials
                else:
                    n, result = int(n), []
                    for k in xrange(n + 1):
                        result.append(C.binomial(n, k)*cls(k)*sym**(n - k))
                    return Add(*result)
            else:
                raise ValueError("Bernoulli numbers are defined only"
                                 " for nonnegative integer indices.")


#----------------------------------------------------------------------------#
#                                                                            #
#                             Bell numbers                                   #
#                                                                            #
#----------------------------------------------------------------------------#

class bell(Function):
    r"""
    Bell numbers / Bell polynomials

    The Bell numbers satisfy `B_0 = 1` and

    .. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.

    They are also given by:

    .. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.

    The Bell polynomials are given by `B_0(x) = 1` and

    .. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).

    The second kind of Bell polynomials (are sometimes called "partial" Bell
    polynomials or incomplete Bell polynomials) are defined as

    .. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
            \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
                \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
                \left(\frac{x_1}{1!} \right)^{j_1}
                \left(\frac{x_2}{2!} \right)^{j_2} \dotsb
                \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.

    * bell(n) gives the `n^{th}` Bell number, `B_n`.
    * bell(n, x) gives the `n^{th}` Bell polynomial, `B_n(x)`.
    * bell(n, k, (x1, x2, ...)) gives Bell polynomials of the second kind,
      `B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.

    Notes
    =====

    Not to be confused with Bernoulli numbers and Bernoulli polynomials,
    which use the same notation.

    Examples
    ========

    >>> from sympy import bell, Symbol, symbols

    >>> [bell(n) for n in range(11)]
    [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
    >>> bell(30)
    846749014511809332450147
    >>> bell(4, Symbol('t'))
    t**4 + 6*t**3 + 7*t**2 + t
    >>> bell(6, 2, symbols('x:6')[1:])
    6*x1*x5 + 15*x2*x4 + 10*x3**2

    References
    ==========

    * http://en.wikipedia.org/wiki/Bell_number
    * http://mathworld.wolfram.com/BellNumber.html
    * http://mathworld.wolfram.com/BellPolynomial.html

    See Also
    ========

    euler, bernoulli
    """

    @staticmethod
    @recurrence_memo([1, 1])
    def _bell(n, prev):
        s = 1
        a = 1
        for k in xrange(1, n):
            a = a * (n - k) // k
            s += a * prev[k]
        return s

    @staticmethod
    @recurrence_memo([S.One, _sym])
    def _bell_poly(n, prev):
        s = 1
        a = 1
        for k in xrange(2, n + 1):
            a = a * (n - k + 1) // (k - 1)
            s += a * prev[k - 1]
        return expand_mul(_sym * s)

    @staticmethod
    def _bell_incomplete_poly(n, k, symbols):
        r"""
        The second kind of Bell polynomials (incomplete Bell polynomials).

        Calculated by recurrence formula:

        .. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
                \sum_{m=1}^{n-k+1}
                \x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})

        where
            B_{0,0} = 1;
            B_{n,0} = 0; for n>=1
            B_{0,k} = 0; for k>=1

        """
        if (n == 0) and (k == 0):
            return S.One
        elif (n == 0) or (k == 0):
            return S.Zero
        s = S.Zero
        a = S.One
        for m in xrange(1, n - k + 2):
            s += a * bell._bell_incomplete_poly(
                n - m, k - 1, symbols) * symbols[m - 1]
            a = a * (n - m) / m
        return expand_mul(s)

    @classmethod
    def eval(cls, n, k_sym=None, symbols=None):
        if n.is_Integer and n.is_nonnegative:
            if k_sym is None:
                return Integer(cls._bell(int(n)))
            elif symbols is None:
                return cls._bell_poly(int(n)).subs(_sym, k_sym)
            else:
                r = cls._bell_incomplete_poly(int(n), int(k_sym), symbols)
                return r

#----------------------------------------------------------------------------#
#                                                                            #
#                           Harmonic numbers                                 #
#                                                                            #
#----------------------------------------------------------------------------#


class harmonic(Function):
    r"""
    Harmonic numbers

    The nth harmonic number is given by 1 + 1/2 + 1/3 + ... + 1/n.

    More generally::

                   n
                  ___
                 \       -m
          H    =  )     k   .
           n,m   /___
                 k = 1

    As n -> oo, H_{n,m} -> zeta(m) (the Riemann zeta function)

    * harmonic(n) gives the nth harmonic number, H_n

    * harmonic(n, m) gives the nth generalized harmonic number
      of order m, H_{n,m}, where harmonic(n) == harmonic(n, 1)

    Examples
    ========

    >>> from sympy import harmonic, oo

    >>> [harmonic(n) for n in range(6)]
    [0, 1, 3/2, 11/6, 25/12, 137/60]
    >>> [harmonic(n, 2) for n in range(6)]
    [0, 1, 5/4, 49/36, 205/144, 5269/3600]
    >>> harmonic(oo, 2)
    pi**2/6

    We can rewrite harmonic numbers in terms of polygamma functions:

    >>> from sympy import Symbol, digamma, polygamma
    >>> n = Symbol("n")
    >>> m = Symbol("m")

    >>> harmonic(n).rewrite(digamma)
    polygamma(0, n + 1) + EulerGamma

    >>> harmonic(n).rewrite(polygamma)
    polygamma(0, n + 1) + EulerGamma

    >>> harmonic(n,3).rewrite(polygamma)
    polygamma(2, n + 1)/2 - polygamma(2, 1)/2

    >>> harmonic(n,m).rewrite(polygamma)
    (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/(m - 1)!

    Integer offsets in the argument can be pulled out:

    >>> from sympy import expand_func

    >>> expand_func(harmonic(n+4))
    harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)

    >>> expand_func(harmonic(n-4))
    harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n

    Some limits can be computed as well:

    >>> from sympy import limit, oo

    >>> limit(harmonic(n), n, oo)
    oo

    >>> limit(harmonic(n, 2), n, oo)
    pi**2/6

    >>> limit(harmonic(n, 3), n, oo)
    -polygamma(2, 1)/2

    >>> limit(harmonic(m, n), m, oo)
    zeta(n)

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Harmonic_number
    .. [2] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
    .. [3] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/
    """

    # Generate one memoized Harmonic number-generating function for each
    # order and store it in a dictionary
    _functions = {}

    nargs = (1, 2)

    @classmethod
    def eval(cls, n, m=None):
        if m is None:
            m = S.One
        if n == oo:
            return C.zeta(m)
        if n.is_Integer and n.is_nonnegative and m.is_Integer:
            if n == 0:
                return S.Zero
            if not m in cls._functions:
                @recurrence_memo([0])
                def f(n, prev):
                    return prev[-1] + S.One / n**m
                cls._functions[m] = f
            return cls._functions[m](int(n))

    def _eval_rewrite_as_polygamma(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return S.NegativeOne**m/factorial(m - 1) * (polygamma(m - 1, 1) - polygamma(m - 1, n + 1))

    def _eval_rewrite_as_digamma(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return self.rewrite(polygamma)

    def _eval_rewrite_as_trigamma(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return self.rewrite(polygamma)

    def _eval_expand_func(self, **hints):
        n = self.args[0]
        m = self.args[1] if len(self.args) == 2 else 1

        if m == S.One:
            if n.is_Add:
                off = n.args[0]
                nnew = n - off
                if off.is_Integer and off.is_positive:
                    result = [S.One/(nnew + i) for i in xrange(off, 0, -1)] + [harmonic(nnew)]
                    return Add(*result)
                elif off.is_Integer and off.is_negative:
                    result = [-S.One/(nnew + i) for i in xrange(0, off, -1)] + [harmonic(nnew)]
                    return Add(*result)

        return self

    def _eval_rewrite_as_tractable(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return self.rewrite(polygamma).rewrite("tractable", deep=True)


#----------------------------------------------------------------------------#
#                                                                            #
#                           Euler numbers                                    #
#                                                                            #
#----------------------------------------------------------------------------#


class euler(Function):
    r"""
    Euler numbers

    The euler numbers are given by::

                  2*n+1   k
                   ___   ___            j          2*n+1
                  \     \     / k \ (-1)  * (k-2*j)
          E   = I  )     )    |   | --------------------
           2n     /___  /___  \ j /      k    k
                  k = 1 j = 0           2  * I  * k

          E     = 0
           2n+1

    * euler(n) gives the n-th Euler number, E_n

    Examples
    ========

    >>> from sympy import Symbol, euler
    >>> [euler(n) for n in range(10)]
    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
    >>> n = Symbol("n")
    >>> euler(n+2*n)
    euler(3*n)

    References
    ==========

    * http://en.wikipedia.org/wiki/Euler_numbers
    * http://mathworld.wolfram.com/EulerNumber.html
    * http://en.wikipedia.org/wiki/Alternating_permutation
    * http://mathworld.wolfram.com/AlternatingPermutation.html

    See Also
    ========

    bernoulli, bell
    """

    nargs = 1

    @classmethod
    def eval(cls, m, evaluate=True):
        if not evaluate:
            return
        if m.is_odd:
            return S.Zero
        if m.is_Integer and m.is_nonnegative:
            from sympy.mpmath import mp
            m = m._to_mpmath(mp.prec)
            res = mp.eulernum(m, exact=True)
            return Integer(res)

    def _eval_rewrite_as_Sum(self, arg):
        if arg.is_even:
            k = C.Dummy("k", integer=True)
            j = C.Dummy("j", integer=True)
            n = self.args[0] / 2
            Em = (S.ImaginaryUnit * C.Sum( C.Sum( C.binomial(k, j) * ((-1)**j * (k - 2*j)**(2*n + 1)) /
                  (2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))

            return Em

    def _eval_evalf(self, prec):
        m = self.args[0]

        if m.is_Integer and m.is_nonnegative:
            from sympy.mpmath import mp
            from sympy import Expr
            m = m._to_mpmath(prec)
            oprec = mp.prec
            mp.prec = prec
            res = mp.eulernum(m)
            mp.prec = oprec
            return Expr._from_mpmath(res, prec)

#----------------------------------------------------------------------------#
#                                                                            #
#                           Catalan numbers                                  #
#                                                                            #
#----------------------------------------------------------------------------#


class catalan(Function):
    r"""
    Catalan numbers

    The n-th catalan number is given by::

                 1   / 2*n \
          C  = ----- |     |
           n   n + 1 \  n  /

    * catalan(n) gives the n-th Catalan number, C_n

    Examples
    ========

    >>> from sympy import (Symbol, binomial, gamma, hyper, polygamma,
    ...             catalan, diff, combsimp, Rational, I)

    >>> [ catalan(i) for i in range(1,10) ]
    [1, 2, 5, 14, 42, 132, 429, 1430, 4862]

    >>> n = Symbol("n", integer=True)

    >>> catalan(n)
    catalan(n)

    Catalan numbers can be transformed into several other, identical
    expressions involving other mathematical functions

    >>> catalan(n).rewrite(binomial)
    binomial(2*n, n)/(n + 1)

    >>> catalan(n).rewrite(gamma)
    4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))

    >>> catalan(n).rewrite(hyper)
    hyper((-n + 1, -n), (2,), 1)

    For some non-integer values of n we can get closed form
    expressions by rewriting in terms of gamma functions:

    >>> catalan(Rational(1,2)).rewrite(gamma)
    8/(3*pi)

    We can differentiate the Catalan numbers C(n) interpreted as a
    continuous real funtion in n:

    >>> diff(catalan(n), n)
    (polygamma(0, n + 1/2) - polygamma(0, n + 2) + 2*log(2))*catalan(n)

    As a more advanced example consider the following ratio
    between consecutive numbers:

    >>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))
    2*(2*n + 1)/(n + 2)

    The Catalan numbers can be generalized to complex numbers:

    >>> catalan(I).rewrite(gamma)
    4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))

    and evaluated with arbitrary precision:

    >>> catalan(I).evalf(20)
    0.39764993382373624267 - 0.020884341620842555705*I

    References
    ==========

    * http://en.wikipedia.org/wiki/Catalan_number
    * http://mathworld.wolfram.com/CatalanNumber.html
    * http://geometer.org/mathcircles/catalan.pdf

    See Also
    ========

    sympy.functions.combinatorial.factorials.binomial
    """

    @classmethod
    def eval(cls, n, evaluate=True):
        if n.is_Integer and n.is_nonnegative:
            return 4**n*C.gamma(n + S.Half)/(C.gamma(S.Half)*C.gamma(n + 2))

    def fdiff(self, argindex=1):
        n = self.args[0]
        return catalan(n)*(C.polygamma(0, n + Rational(1, 2)) - C.polygamma(0, n + 2) + C.log(4))

    def _eval_rewrite_as_binomial(self, n):
        return C.binomial(2*n, n)/(n + 1)

    def _eval_rewrite_as_gamma(self, n):
        # The gamma function allows to generalize Catalan numbers to complex n
        return 4**n*C.gamma(n + S.Half)/(C.gamma(S.Half)*C.gamma(n + 2))

    def _eval_rewrite_as_hyper(self, n):
        return C.hyper([1 - n, -n], [2], 1)

    def _eval_evalf(self, prec):
        return self.rewrite(C.gamma).evalf(prec)

#######################################################################
###
### Functions for enumerating partitions, permutations and combinations
###
#######################################################################


class _MultisetHistogram(tuple):
    pass


_N = -1
_ITEMS = -2
_M = slice(None, _ITEMS)


def _multiset_histogram(n):
    """Return tuple used in permutation and combination counting. Input
    is a dictionary giving items with counts as values or a sequence of
    items (which need not be sorted).

    The data is stored in a class deriving from tuple so it is easily
    recognized and so it can be converted easily to a list.
    """
    if type(n) is dict:  # item: count
        if not all(isinstance(v, int) and v >= 0 for v in n.values()):
            raise ValueError
        tot = sum(n.values())
        items = sum(1 for k in n if n[k] > 0)
        return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
    else:
        n = list(n)
        s = set(n)
        if len(s) == len(n):
            n = [1]*len(n)
            n.extend([len(n), len(n)])
            return _MultisetHistogram(n)
        m = dict(zip(s, range(len(s))))
        d = dict(zip(range(len(s)), [0]*len(s)))
        for i in n:
            d[m[i]] += 1
        return _multiset_histogram(d)


def nP(n, k=None, replacement=False):
    """Return the number of permutations of ``n`` items taken ``k`` at a time.

    Possible values for ``n``::
        integer - set of length ``n``
        sequence - converted to a multiset internally
        multiset - {element: multiplicity}

    If ``k`` is None then the total of all permutations of length 0
    through the number of items represented by ``n`` will be returned.

    If ``replacement`` is True then a given item can appear more than once
    in the ``k`` items. (For example, for 'ab' permutations of 2 would
    include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in
    ``n`` is ignored when ``replacement`` is True but the total number
    of elements is considered since no element can appear more times than
    the number of elements in ``n``.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import nP
    >>> from sympy.utilities.iterables import multiset_permutations, multiset
    >>> nP(3, 2)
    6
    >>> nP('abc', 2) == nP(multiset('abc'), 2) == 6
    True
    >>> nP('aab', 2)
    3
    >>> nP([1, 2, 2], 2)
    3
    >>> [nP(3, i) for i in range(4)]
    [1, 3, 6, 6]
    >>> nP(3) == sum(_)
    True

    When ``replacement`` is True, each item can have multiplicity
    equal to the length represented by ``n``:

    >>> nP('aabc', replacement=True)
    121
    >>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)]
    [1, 3, 9, 27, 81]
    >>> sum(_)
    121

    References
    ==========

    http://en.wikipedia.org/wiki/Permutation

    See Also
    ========
    sympy.utilities.iterables.multiset_permutations

    """
    try:
        n = as_int(n)
    except ValueError:
        return Integer(_nP(_multiset_histogram(n), k, replacement))
    return Integer(_nP(n, k, replacement))


@cacheit
def _nP(n, k=None, replacement=False):
    from sympy.functions.combinatorial.factorials import factorial
    from sympy.core.mul import prod

    if k == 0:
        return 1
    if isinstance(n, SYMPY_INTS):  # n different items
        # assert n >= 0
        if k is None:
            return sum(_nP(n, i, replacement) for i in range(n + 1))
        elif replacement:
            return n**k
        elif k > n:
            return 0
        elif k == n:
            return factorial(k)
        elif k == 1:
            return n
        else:
            # assert k >= 0
            return _product(n - k + 1, n)
    elif isinstance(n, _MultisetHistogram):
        if k is None:
            return sum(_nP(n, i, replacement) for i in range(n[_N] + 1))
        elif replacement:
            return n[_ITEMS]**k
        elif k == n[_N]:
            return factorial(k)/prod([factorial(i) for i in n[_M] if i > 1])
        elif k > n[_N]:
            return 0
        elif k == 1:
            return n[_ITEMS]
        else:
            # assert k >= 0
            tot = 0
            n = list(n)
            for i in range(len(n[_M])):
                if not n[i]:
                    continue
                n[_N] -= 1
                if n[i] == 1:
                    n[i] = 0
                    n[_ITEMS] -= 1
                    tot += _nP(_MultisetHistogram(n), k - 1)
                    n[_ITEMS] += 1
                    n[i] = 1
                else:
                    n[i] -= 1
                    tot += _nP(_MultisetHistogram(n), k - 1)
                    n[i] += 1
                n[_N] += 1
            return tot


@cacheit
def _AOP_product(n):
    """for n = (m1, m2, .., mk) return the coefficients of the polynomial,
    prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients
    of the product of AOPs (all-one polynomials) or order given in n.  The
    resulting coefficient corresponding to x**r is the number of r-length
    combinations of sum(n) elements with multiplicities given in n.
    The coefficients are given as a default dictionary (so if a query is made
    for a key that is not present, 0 will be returned).

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import _AOP_product
    >>> from sympy.abc import x
    >>> n = (2, 2, 3)  # e.g. aabbccc
    >>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand()
    >>> c = _AOP_product(n); dict(c)
    {0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1}
    >>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)]
    True

    The generating poly used here is the same as that listed in
    http://tinyurl.com/cep849r, but in a refactored form.

    """
    from collections import defaultdict

    n = list(n)
    ord = sum(n)
    need = (ord + 2)//2
    rv = [1]*(n.pop() + 1)
    rv.extend([0]*(need - len(rv)))
    rv = rv[:need]
    while n:
        ni = n.pop()
        N = ni + 1
        was = rv[:]
        for i in range(1, min(N, len(rv))):
            rv[i] += rv[i - 1]
        for i in range(N, need):
            rv[i] += rv[i - 1] - was[i - N]
    rev = list(reversed(rv))
    if ord % 2:
        rv = rv + rev
    else:
        rv[-1:] = rev
    d = defaultdict(int)
    for i in range(len(rv)):
        d[i] = rv[i]
    return d


def nC(n, k=None, replacement=False):
    """Return the number of combinations of ``n`` items taken ``k`` at a time.

    Possible values for ``n``::
        integer - set of length ``n``
        sequence - converted to a multiset internally
        multiset - {element: multiplicity}

    If ``k`` is None then the total of all combinations of length 0
    through the number of items represented in ``n`` will be returned.

    If ``replacement`` is True then a given item can appear more than once
    in the ``k`` items. (For example, for 'ab' sets of 2 would include 'aa',
    'ab', and 'bb'.) The multiplicity of elements in ``n`` is ignored when
    ``replacement`` is True but the total number of elements is considered
    since no element can appear more times than the number of elements in
    ``n``.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import nC
    >>> from sympy.utilities.iterables import multiset_combinations
    >>> nC(3, 2)
    3
    >>> nC('abc', 2)
    3
    >>> nC('aab', 2)
    2

    When ``replacement`` is True, each item can have multiplicity
    equal to the length represented by ``n``:

    >>> nC('aabc', replacement=True)
    35
    >>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)]
    [1, 3, 6, 10, 15]
    >>> sum(_)
    35

    If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k``
    then the total of all combinations of length 0 hrough ``k`` is the
    product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity
    of each item is 1 (i.e., k unique items) then there are 2**k
    combinations. For example, if there are 4 unique items, the total number
    of combinations is 16:

    >>> sum(nC(4, i) for i in range(5))
    16

    References
    ==========

    * http://en.wikipedia.org/wiki/Combination
    * http://tinyurl.com/cep849r

    See Also
    ========
    sympy.utilities.iterables.multiset_combinations
    """
    from sympy.functions.combinatorial.factorials import binomial
    from sympy.core.mul import prod

    if isinstance(n, SYMPY_INTS):
        if k is None:
            if not replacement:
                return 2**n
            return sum(nC(n, i, replacement) for i in range(n + 1))
        assert k >= 0
        if replacement:
            return binomial(n + k - 1, k)
        return binomial(n, k)
    if isinstance(n, _MultisetHistogram):
        N = n[_N]
        if k is None:
            if not replacement:
                return prod(m + 1 for m in n[_M])
            return sum(nC(n, i, replacement) for i in range(N + 1))
        elif replacement:
            return nC(n[_ITEMS], k, replacement)
        # assert k >= 0
        elif k in (1, N - 1):
            return n[_ITEMS]
        elif k in (0, N):
            return 1
        return _AOP_product(tuple(n[_M]))[k]
    else:
        return nC(_multiset_histogram(n), k, replacement)


@cacheit
def _stirling1(n, k):
    if n == k == 0:
        return S.One
    if 0 in (n, k):
        return S.Zero
    n1 = n - 1

    # some special values
    if n == k:
        return S.One
    elif k == 1:
        return factorial(n1)
    elif k == n1:
        return C.binomial(n, 2)
    elif k == n - 2:
        return (3*n - 1)*C.binomial(n, 3)/4
    elif k == n - 3:
        return C.binomial(n, 2)*C.binomial(n, 4)

    # general recurrence
    return n1*_stirling1(n1, k) + _stirling1(n1, k - 1)


@cacheit
def _stirling2(n, k):
    if n == k == 0:
        return S.One
    if 0 in (n, k):
        return S.Zero
    n1 = n - 1

    # some special values
    if k == n1:
        return C.binomial(n, 2)
    elif k == 2:
        return 2**n1 - 1

    # general recurrence
    return k*_stirling2(n1, k) + _stirling2(n1, k - 1)


def stirling(n, k, d=None, kind=2, signed=False):
    """Return Stirling number S(n, k) of the first or second (default) kind.

    The sum of all Stirling numbers of the second kind for k = 1
    through n is bell(n). The recurrence relationship for these numbers
    is::

    {0}       {n}   {0}      {n + 1}     {n}   {  n  }
    { } = 1;  { } = { } = 0; {     } = j*{ } + {     }
    {0}       {0}   {k}      {  k  }     {k}   {k - 1}

    where ``j`` is::
        ``n`` for Stirling numbers of the first kind
        ``-n`` for signed Stirling numbers of the first kind
        ``k`` for Stirling numbers of the second kind

    The first kind of Stirling number counts the number of permutations of
    ``n`` distinct items that have ``k`` cycles; the second kind counts the
    ways in which ``n`` distinct items can be partitioned into ``k`` parts.
    If ``d`` is given, the "reduced Stirling number of the second kind" is
    returned: ``S^{d}(n, k) = S(n - d + 1, k - d + 1)`` with ``n >= k >= d``.
    (This counts the ways to partition ``n`` consecutive integers into
    ``k`` groups with no pairwise difference less than ``d``. See example
    below.)

    To obtain the signed Stirling numbers of the first kind, use keyword
    ``signed=True``. Using this keyword automatically sets ``kind`` to 1.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import stirling, bell
    >>> from sympy.combinatorics import Permutation
    >>> from sympy.utilities.iterables import multiset_partitions, permutations

    First kind (unsigned by default):

    >>> [stirling(6, i, kind=1) for i in range(7)]
    [0, 120, 274, 225, 85, 15, 1]
    >>> perms = list(permutations(range(4)))
    >>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)]
    [0, 6, 11, 6, 1]
    >>> [stirling(4, i, kind=1) for i in range(5)]
    [0, 6, 11, 6, 1]

    First kind (signed):

    >>> [stirling(4, i, signed=True) for i in range(5)]
    [0, -6, 11, -6, 1]

    Second kind:

    >>> [stirling(10, i) for i in range(12)]
    [0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0]
    >>> sum(_) == bell(10)
    True
    >>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2)
    True

    Reduced second kind:

    >>> from sympy import subsets, oo
    >>> def delta(p):
    ...    if len(p) == 1:
    ...        return oo
    ...    return min(abs(i[0] - i[1]) for i in subsets(p, 2))
    >>> parts = multiset_partitions(range(5), 3)
    >>> d = 2
    >>> sum(1 for p in parts if all(delta(i) >= d for i in p))
    7
    >>> stirling(5, 3, 2)
    7

    References
    ==========

    * http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
    * http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

    See Also
    ========
    sympy.utilities.iterables.multiset_partitions

    """
    # TODO: make this a class like bell()

    n = as_int(n)
    k = as_int(k)
    if n < 0:
        raise ValueError('n must be nonnegative')
    if k > n:
        return S.Zero
    if d:
        # assert k >= d
        # kind is ignored -- only kind=2 is supported
        return _stirling2(n - d + 1, k - d + 1)
    elif signed:
        # kind is ignored -- only kind=1 is supported
        return (-1)**(n - k)*_stirling1(n, k)

    if kind == 1:
        return _stirling1(n, k)
    elif kind == 2:
        return _stirling2(n, k)
    else:
        raise ValueError('kind must be 1 or 2, not %s' % k)


@cacheit
def _nT(n, k):
    """Return the partitions of ``n`` items into ``k`` parts. This
    is used by ``nT`` for the case when ``n`` is an integer."""
    if k == 0:
        return 1 if k == n else 0
    return sum(_nT(n - k, j) for j in range(min(k, n - k) + 1))


def nT(n, k=None):
    """Return the number of ``k``-sized partitions of ``n`` items.

    Possible values for ``n``::
        integer - ``n`` identical items
        sequence - converted to a multiset internally
        multiset - {element: multiplicity}

    Note: the convention for ``nT`` is different than that of ``nC`` and``nP`` in that
    here an integer indicates ``n`` *identical* items instead of a set of
    length ``n``; this is in keepng with the ``partitions`` function which
    treats its integer-``n`` input like a list of ``n`` 1s. One can use
    ``range(n)`` for ``n`` to indicate ``n`` distinct items.

    If ``k`` is None then the total number of ways to partition the elements
    represented in ``n`` will be returned.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import nT

    Partitions of the given multiset:

    >>> [nT('aabbc', i) for i in range(1, 7)]
    [1, 8, 11, 5, 1, 0]
    >>> nT('aabbc') == sum(_)
    True

    (TODO The following can be activated with >>> when
    taocp_multiset_permutation is in place.)
    >> [nT("mississippi", i) for i in range(1, 12)]
    [1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1]

    Partitions when all items are identical:

    >>> [nT(5, i) for i in range(1, 6)]
    [1, 2, 2, 1, 1]
    >>> nT('1'*5) == sum(_)
    True

    When all items are different:

    >>> [nT(range(5), i) for i in range(1, 6)]
    [1, 15, 25, 10, 1]
    >>> nT(range(5)) == sum(_)
    True

    References
    ==========

    * http://undergraduate.csse.uwa.edu.au/units/CITS7209/partition.pdf

    See Also
    ========
    sympy.utilities.iterables.partitions
    sympy.utilities.iterables.multiset_partitions

    """
    from sympy.utilities.iterables import multiset_partitions

    if isinstance(n, SYMPY_INTS):
        # assert n >= 0
        # all the same
        if k is None:
            return sum(_nT(n, k) for k in range(1, n + 1))
        return _nT(n, k)
    if not isinstance(n, _MultisetHistogram):
        try:
            # if n contains hashable items there is some
            # quick handling that can be done
            u = len(set(n))
            if u == 1:
                return nT(len(n), k)
            elif u == len(n):
                n = range(u)
            raise TypeError
        except TypeError:
            n = _multiset_histogram(n)
    N = n[_N]
    if k is None and N == 1:
        return 1
    if k in (1, N):
        return 1
    if k == 2 or N == 2 and k is None:
        m, r = divmod(N, 2)
        rv = sum(nC(n, i) for i in range(1, m + 1))
        if not r:
            rv -= nC(n, m)//2
        if k is None:
            rv += 1  # for k == 1
        return rv
    if N == n[_ITEMS]:
        # all distinct
        if k is None:
            return bell(N)
        return stirling(N, k)
    if k is None:
        return sum(nT(n, k) for k in range(1, N + 1))
    tot = 0
    for p in multiset_partitions(
            [i for i, j in enumerate(n[_M]) for ii in range(j)]):
        tot += len(p) == k
    return tot
<commit_msg>Rewrite of harmonic numbers into summation<commit_after>"""
This module implements some special functions that commonly appear in
combinatorial contexts (e.g. in power series); in particular,
sequences of rational numbers such as Bernoulli and Fibonacci numbers.

Factorials, binomial coefficients and related functions are located in
the separate 'factorials' module.
"""

from sympy.core.function import Function, expand_mul
from sympy.core import S, Symbol, Rational, oo, Integer, C, Add, Dummy
from sympy.core.compatibility import as_int, SYMPY_INTS
from sympy.core.cache import cacheit
from sympy.functions.combinatorial.factorials import factorial

from sympy.mpmath import bernfrac
from sympy.mpmath.libmp import ifib as _ifib


def _product(a, b):
    p = 1
    for k in xrange(a, b + 1):
        p *= k
    return p

from sympy.utilities.memoization import recurrence_memo


# Dummy symbol used for computing polynomial sequences
_sym = Symbol('x')
_symbols = Function('x')


#----------------------------------------------------------------------------#
#                                                                            #
#                           Fibonacci numbers                                #
#                                                                            #
#----------------------------------------------------------------------------#

class fibonacci(Function):
    """
    Fibonacci numbers / Fibonacci polynomials

    The Fibonacci numbers are the integer sequence defined by the
    initial terms F_0 = 0, F_1 = 1 and the two-term recurrence
    relation F_n = F_{n-1} + F_{n-2}.

    The Fibonacci polynomials are defined by F_1(x) = 1,
    F_2(x) = x, and F_n(x) = x*F_{n-1}(x) + F_{n-2}(x) for n > 2.
    For all positive integers n, F_n(1) = F_n.

    * fibonacci(n) gives the nth Fibonacci number, F_n
    * fibonacci(n, x) gives the nth Fibonacci polynomial in x, F_n(x)

    Examples
    ========

    >>> from sympy import fibonacci, Symbol

    >>> [fibonacci(x) for x in range(11)]
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
    >>> fibonacci(5, Symbol('t'))
    t**4 + 3*t**2 + 1

    References
    ==========

    * http://en.wikipedia.org/wiki/Fibonacci_number
    * http://mathworld.wolfram.com/FibonacciNumber.html

    See Also
    ========

    lucas
    """

    @staticmethod
    def _fib(n):
        return _ifib(n)

    @staticmethod
    @recurrence_memo([None, S.One, _sym])
    def _fibpoly(n, prev):
        return (prev[-2] + _sym*prev[-1]).expand()

    @classmethod
    def eval(cls, n, sym=None):
        if n.is_Integer:
            n = int(n)
            if n < 0:
                return S.NegativeOne**(n + 1) * fibonacci(-n)
            if sym is None:
                return Integer(cls._fib(n))
            else:
                if n < 1:
                    raise ValueError("Fibonacci polynomials are defined "
                       "only for positive integer indices.")
                return cls._fibpoly(n).subs(_sym, sym)


class lucas(Function):
    """
    Lucas numbers

    Lucas numbers satisfy a recurrence relation similar to that of
    the Fibonacci sequence, in which each term is the sum of the
    preceding two. They are generated by choosing the initial
    values L_0 = 2 and L_1 = 1.

    * lucas(n) gives the nth Lucas number

    Examples
    ========

    >>> from sympy import lucas

    >>> [lucas(x) for x in range(11)]
    [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]

    References
    ==========

    * http://en.wikipedia.org/wiki/Lucas_number

    See Also
    ========

    fibonacci
    """

    @classmethod
    def eval(cls, n):
        if n.is_Integer:
            return fibonacci(n + 1) + fibonacci(n - 1)


#----------------------------------------------------------------------------#
#                                                                            #
#                           Bernoulli numbers                                #
#                                                                            #
#----------------------------------------------------------------------------#

class bernoulli(Function):
    r"""
    Bernoulli numbers / Bernoulli polynomials

    The Bernoulli numbers are a sequence of rational numbers
    defined by B_0 = 1 and the recursive relation (n > 0)::

                n
               ___
              \      / n + 1 \
          0 =  )     |       | * B .
              /___   \   k   /    k
              k = 0

    They are also commonly defined by their exponential generating
    function, which is x/(exp(x) - 1). For odd indices > 1, the
    Bernoulli numbers are zero.

    The Bernoulli polynomials satisfy the analogous formula::

                    n
                   ___
                  \      / n \         n-k
          B (x) =  )     |   | * B  * x   .
           n      /___   \ k /    k
                  k = 0

    Bernoulli numbers and Bernoulli polynomials are related as
    B_n(0) = B_n.

    We compute Bernoulli numbers using Ramanujan's formula::

                                   / n + 3 \
          B   =  (A(n) - S(n))  /  |       |
           n                       \   n   /

    where A(n) = (n+3)/3 when n = 0 or 2 (mod 6), A(n) = -(n+3)/6
    when n = 4 (mod 6), and::

                 [n/6]
                  ___
                 \      /  n + 3  \
          S(n) =  )     |         | * B
                 /___   \ n - 6*k /    n-6*k
                 k = 1

    This formula is similar to the sum given in the definition, but
    cuts 2/3 of the terms. For Bernoulli polynomials, we use the
    formula in the definition.

    * bernoulli(n) gives the nth Bernoulli number, B_n
    * bernoulli(n, x) gives the nth Bernoulli polynomial in x, B_n(x)

    Examples
    ========

    >>> from sympy import bernoulli

    >>> [bernoulli(n) for n in range(11)]
    [1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
    >>> bernoulli(1000001)
    0

    References
    ==========

    * http://en.wikipedia.org/wiki/Bernoulli_number
    * http://en.wikipedia.org/wiki/Bernoulli_polynomial

    See Also
    ========

    euler, bell
    """

    # Calculates B_n for positive even n
    @staticmethod
    def _calc_bernoulli(n):
        s = 0
        a = int(C.binomial(n + 3, n - 6))
        for j in xrange(1, n//6 + 1):
            s += a * bernoulli(n - 6*j)
            # Avoid computing each binomial coefficient from scratch
            a *= _product(n - 6 - 6*j + 1, n - 6*j)
            a //= _product(6*j + 4, 6*j + 9)
        if n % 6 == 4:
            s = -Rational(n + 3, 6) - s
        else:
            s = Rational(n + 3, 3) - s
        return s / C.binomial(n + 3, n)

    # We implement a specialized memoization scheme to handle each
    # case modulo 6 separately
    _cache = {0: S.One, 2: Rational(1, 6), 4: Rational(-1, 30)}
    _highest = {0: 0, 2: 2, 4: 4}

    @classmethod
    def eval(cls, n, sym=None):
        if n.is_Number:
            if n.is_Integer and n.is_nonnegative:
                if n is S.Zero:
                    return S.One
                elif n is S.One:
                    if sym is None:
                        return -S.Half
                    else:
                        return sym - S.Half
                # Bernoulli numbers
                elif sym is None:
                    if n.is_odd:
                        return S.Zero
                    n = int(n)
                    # Use mpmath for enormous Bernoulli numbers
                    if n > 500:
                        p, q = bernfrac(n)
                        return Rational(int(p), int(q))
                    case = n % 6
                    highest_cached = cls._highest[case]
                    if n <= highest_cached:
                        return cls._cache[n]
                    # To avoid excessive recursion when, say, bernoulli(1000) is
                    # requested, calculate and cache the entire sequence ... B_988,
                    # B_994, B_1000 in increasing order
                    for i in xrange(highest_cached + 6, n + 6, 6):
                        b = cls._calc_bernoulli(i)
                        cls._cache[i] = b
                        cls._highest[case] = i
                    return b
                # Bernoulli polynomials
                else:
                    n, result = int(n), []
                    for k in xrange(n + 1):
                        result.append(C.binomial(n, k)*cls(k)*sym**(n - k))
                    return Add(*result)
            else:
                raise ValueError("Bernoulli numbers are defined only"
                                 " for nonnegative integer indices.")


#----------------------------------------------------------------------------#
#                                                                            #
#                             Bell numbers                                   #
#                                                                            #
#----------------------------------------------------------------------------#

class bell(Function):
    r"""
    Bell numbers / Bell polynomials

    The Bell numbers satisfy `B_0 = 1` and

    .. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.

    They are also given by:

    .. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.

    The Bell polynomials are given by `B_0(x) = 1` and

    .. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).

    The second kind of Bell polynomials (are sometimes called "partial" Bell
    polynomials or incomplete Bell polynomials) are defined as

    .. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
            \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
                \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
                \left(\frac{x_1}{1!} \right)^{j_1}
                \left(\frac{x_2}{2!} \right)^{j_2} \dotsb
                \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.

    * bell(n) gives the `n^{th}` Bell number, `B_n`.
    * bell(n, x) gives the `n^{th}` Bell polynomial, `B_n(x)`.
    * bell(n, k, (x1, x2, ...)) gives Bell polynomials of the second kind,
      `B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.

    Notes
    =====

    Not to be confused with Bernoulli numbers and Bernoulli polynomials,
    which use the same notation.

    Examples
    ========

    >>> from sympy import bell, Symbol, symbols

    >>> [bell(n) for n in range(11)]
    [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
    >>> bell(30)
    846749014511809332450147
    >>> bell(4, Symbol('t'))
    t**4 + 6*t**3 + 7*t**2 + t
    >>> bell(6, 2, symbols('x:6')[1:])
    6*x1*x5 + 15*x2*x4 + 10*x3**2

    References
    ==========

    * http://en.wikipedia.org/wiki/Bell_number
    * http://mathworld.wolfram.com/BellNumber.html
    * http://mathworld.wolfram.com/BellPolynomial.html

    See Also
    ========

    euler, bernoulli
    """

    @staticmethod
    @recurrence_memo([1, 1])
    def _bell(n, prev):
        s = 1
        a = 1
        for k in xrange(1, n):
            a = a * (n - k) // k
            s += a * prev[k]
        return s

    @staticmethod
    @recurrence_memo([S.One, _sym])
    def _bell_poly(n, prev):
        s = 1
        a = 1
        for k in xrange(2, n + 1):
            a = a * (n - k + 1) // (k - 1)
            s += a * prev[k - 1]
        return expand_mul(_sym * s)

    @staticmethod
    def _bell_incomplete_poly(n, k, symbols):
        r"""
        The second kind of Bell polynomials (incomplete Bell polynomials).

        Calculated by recurrence formula:

        .. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
                \sum_{m=1}^{n-k+1}
                \x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})

        where
            B_{0,0} = 1;
            B_{n,0} = 0; for n>=1
            B_{0,k} = 0; for k>=1

        """
        if (n == 0) and (k == 0):
            return S.One
        elif (n == 0) or (k == 0):
            return S.Zero
        s = S.Zero
        a = S.One
        for m in xrange(1, n - k + 2):
            s += a * bell._bell_incomplete_poly(
                n - m, k - 1, symbols) * symbols[m - 1]
            a = a * (n - m) / m
        return expand_mul(s)

    @classmethod
    def eval(cls, n, k_sym=None, symbols=None):
        if n.is_Integer and n.is_nonnegative:
            if k_sym is None:
                return Integer(cls._bell(int(n)))
            elif symbols is None:
                return cls._bell_poly(int(n)).subs(_sym, k_sym)
            else:
                r = cls._bell_incomplete_poly(int(n), int(k_sym), symbols)
                return r

#----------------------------------------------------------------------------#
#                                                                            #
#                           Harmonic numbers                                 #
#                                                                            #
#----------------------------------------------------------------------------#


class harmonic(Function):
    r"""
    Harmonic numbers

    The nth harmonic number is given by 1 + 1/2 + 1/3 + ... + 1/n.

    More generally::

                   n
                  ___
                 \       -m
          H    =  )     k   .
           n,m   /___
                 k = 1

    As n -> oo, H_{n,m} -> zeta(m) (the Riemann zeta function)

    * harmonic(n) gives the nth harmonic number, H_n

    * harmonic(n, m) gives the nth generalized harmonic number
      of order m, H_{n,m}, where harmonic(n) == harmonic(n, 1)

    Examples
    ========

    >>> from sympy import harmonic, oo

    >>> [harmonic(n) for n in range(6)]
    [0, 1, 3/2, 11/6, 25/12, 137/60]
    >>> [harmonic(n, 2) for n in range(6)]
    [0, 1, 5/4, 49/36, 205/144, 5269/3600]
    >>> harmonic(oo, 2)
    pi**2/6

    We can rewrite harmonic numbers in terms of polygamma functions:

    >>> from sympy import Symbol, digamma, polygamma
    >>> n = Symbol("n")
    >>> m = Symbol("m")

    >>> harmonic(n).rewrite(digamma)
    polygamma(0, n + 1) + EulerGamma

    >>> harmonic(n).rewrite(polygamma)
    polygamma(0, n + 1) + EulerGamma

    >>> harmonic(n,3).rewrite(polygamma)
    polygamma(2, n + 1)/2 - polygamma(2, 1)/2

    >>> harmonic(n,m).rewrite(polygamma)
    (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/(m - 1)!

    Integer offsets in the argument can be pulled out:

    >>> from sympy import expand_func

    >>> expand_func(harmonic(n+4))
    harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)

    >>> expand_func(harmonic(n-4))
    harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n

    Some limits can be computed as well:

    >>> from sympy import limit, oo

    >>> limit(harmonic(n), n, oo)
    oo

    >>> limit(harmonic(n, 2), n, oo)
    pi**2/6

    >>> limit(harmonic(n, 3), n, oo)
    -polygamma(2, 1)/2

    >>> limit(harmonic(m, n), m, oo)
    zeta(n)

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Harmonic_number
    .. [2] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
    .. [3] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/
    """

    # Generate one memoized Harmonic number-generating function for each
    # order and store it in a dictionary
    _functions = {}

    nargs = (1, 2)

    @classmethod
    def eval(cls, n, m=None):
        if m is None:
            m = S.One
        if n == oo:
            return C.zeta(m)
        if n.is_Integer and n.is_nonnegative and m.is_Integer:
            if n == 0:
                return S.Zero
            if not m in cls._functions:
                @recurrence_memo([0])
                def f(n, prev):
                    return prev[-1] + S.One / n**m
                cls._functions[m] = f
            return cls._functions[m](int(n))

    def _eval_rewrite_as_polygamma(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return S.NegativeOne**m/factorial(m - 1) * (polygamma(m - 1, 1) - polygamma(m - 1, n + 1))

    def _eval_rewrite_as_digamma(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return self.rewrite(polygamma)

    def _eval_rewrite_as_trigamma(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return self.rewrite(polygamma)

    def _eval_rewrite_as_Sum(self, n, m=None):
        k = C.Dummy("k", integer=True)
        if m is None:
            m = S.One
        return C.Sum(k**(-m), (k, 1, n))

    def _eval_expand_func(self, **hints):
        n = self.args[0]
        m = self.args[1] if len(self.args) == 2 else 1

        if m == S.One:
            if n.is_Add:
                off = n.args[0]
                nnew = n - off
                if off.is_Integer and off.is_positive:
                    result = [S.One/(nnew + i) for i in xrange(off, 0, -1)] + [harmonic(nnew)]
                    return Add(*result)
                elif off.is_Integer and off.is_negative:
                    result = [-S.One/(nnew + i) for i in xrange(0, off, -1)] + [harmonic(nnew)]
                    return Add(*result)

        return self

    def _eval_rewrite_as_tractable(self, n, m=1):
        from sympy.functions.special.gamma_functions import polygamma
        return self.rewrite(polygamma).rewrite("tractable", deep=True)


#----------------------------------------------------------------------------#
#                                                                            #
#                           Euler numbers                                    #
#                                                                            #
#----------------------------------------------------------------------------#


class euler(Function):
    r"""
    Euler numbers

    The euler numbers are given by::

                  2*n+1   k
                   ___   ___            j          2*n+1
                  \     \     / k \ (-1)  * (k-2*j)
          E   = I  )     )    |   | --------------------
           2n     /___  /___  \ j /      k    k
                  k = 1 j = 0           2  * I  * k

          E     = 0
           2n+1

    * euler(n) gives the n-th Euler number, E_n

    Examples
    ========

    >>> from sympy import Symbol, euler
    >>> [euler(n) for n in range(10)]
    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
    >>> n = Symbol("n")
    >>> euler(n+2*n)
    euler(3*n)

    References
    ==========

    * http://en.wikipedia.org/wiki/Euler_numbers
    * http://mathworld.wolfram.com/EulerNumber.html
    * http://en.wikipedia.org/wiki/Alternating_permutation
    * http://mathworld.wolfram.com/AlternatingPermutation.html

    See Also
    ========

    bernoulli, bell
    """

    nargs = 1

    @classmethod
    def eval(cls, m, evaluate=True):
        if not evaluate:
            return
        if m.is_odd:
            return S.Zero
        if m.is_Integer and m.is_nonnegative:
            from sympy.mpmath import mp
            m = m._to_mpmath(mp.prec)
            res = mp.eulernum(m, exact=True)
            return Integer(res)

    def _eval_rewrite_as_Sum(self, arg):
        if arg.is_even:
            k = C.Dummy("k", integer=True)
            j = C.Dummy("j", integer=True)
            n = self.args[0] / 2
            Em = (S.ImaginaryUnit * C.Sum( C.Sum( C.binomial(k, j) * ((-1)**j * (k - 2*j)**(2*n + 1)) /
                  (2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))

            return Em

    def _eval_evalf(self, prec):
        m = self.args[0]

        if m.is_Integer and m.is_nonnegative:
            from sympy.mpmath import mp
            from sympy import Expr
            m = m._to_mpmath(prec)
            oprec = mp.prec
            mp.prec = prec
            res = mp.eulernum(m)
            mp.prec = oprec
            return Expr._from_mpmath(res, prec)

#----------------------------------------------------------------------------#
#                                                                            #
#                           Catalan numbers                                  #
#                                                                            #
#----------------------------------------------------------------------------#


class catalan(Function):
    r"""
    Catalan numbers

    The n-th catalan number is given by::

                 1   / 2*n \
          C  = ----- |     |
           n   n + 1 \  n  /

    * catalan(n) gives the n-th Catalan number, C_n

    Examples
    ========

    >>> from sympy import (Symbol, binomial, gamma, hyper, polygamma,
    ...             catalan, diff, combsimp, Rational, I)

    >>> [ catalan(i) for i in range(1,10) ]
    [1, 2, 5, 14, 42, 132, 429, 1430, 4862]

    >>> n = Symbol("n", integer=True)

    >>> catalan(n)
    catalan(n)

    Catalan numbers can be transformed into several other, identical
    expressions involving other mathematical functions

    >>> catalan(n).rewrite(binomial)
    binomial(2*n, n)/(n + 1)

    >>> catalan(n).rewrite(gamma)
    4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))

    >>> catalan(n).rewrite(hyper)
    hyper((-n + 1, -n), (2,), 1)

    For some non-integer values of n we can get closed form
    expressions by rewriting in terms of gamma functions:

    >>> catalan(Rational(1,2)).rewrite(gamma)
    8/(3*pi)

    We can differentiate the Catalan numbers C(n) interpreted as a
    continuous real funtion in n:

    >>> diff(catalan(n), n)
    (polygamma(0, n + 1/2) - polygamma(0, n + 2) + 2*log(2))*catalan(n)

    As a more advanced example consider the following ratio
    between consecutive numbers:

    >>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))
    2*(2*n + 1)/(n + 2)

    The Catalan numbers can be generalized to complex numbers:

    >>> catalan(I).rewrite(gamma)
    4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))

    and evaluated with arbitrary precision:

    >>> catalan(I).evalf(20)
    0.39764993382373624267 - 0.020884341620842555705*I

    References
    ==========

    * http://en.wikipedia.org/wiki/Catalan_number
    * http://mathworld.wolfram.com/CatalanNumber.html
    * http://geometer.org/mathcircles/catalan.pdf

    See Also
    ========

    sympy.functions.combinatorial.factorials.binomial
    """

    @classmethod
    def eval(cls, n, evaluate=True):
        if n.is_Integer and n.is_nonnegative:
            return 4**n*C.gamma(n + S.Half)/(C.gamma(S.Half)*C.gamma(n + 2))

    def fdiff(self, argindex=1):
        n = self.args[0]
        return catalan(n)*(C.polygamma(0, n + Rational(1, 2)) - C.polygamma(0, n + 2) + C.log(4))

    def _eval_rewrite_as_binomial(self, n):
        return C.binomial(2*n, n)/(n + 1)

    def _eval_rewrite_as_gamma(self, n):
        # The gamma function allows to generalize Catalan numbers to complex n
        return 4**n*C.gamma(n + S.Half)/(C.gamma(S.Half)*C.gamma(n + 2))

    def _eval_rewrite_as_hyper(self, n):
        return C.hyper([1 - n, -n], [2], 1)

    def _eval_evalf(self, prec):
        return self.rewrite(C.gamma).evalf(prec)

#######################################################################
###
### Functions for enumerating partitions, permutations and combinations
###
#######################################################################


class _MultisetHistogram(tuple):
    pass


_N = -1
_ITEMS = -2
_M = slice(None, _ITEMS)


def _multiset_histogram(n):
    """Return tuple used in permutation and combination counting. Input
    is a dictionary giving items with counts as values or a sequence of
    items (which need not be sorted).

    The data is stored in a class deriving from tuple so it is easily
    recognized and so it can be converted easily to a list.
    """
    if type(n) is dict:  # item: count
        if not all(isinstance(v, int) and v >= 0 for v in n.values()):
            raise ValueError
        tot = sum(n.values())
        items = sum(1 for k in n if n[k] > 0)
        return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
    else:
        n = list(n)
        s = set(n)
        if len(s) == len(n):
            n = [1]*len(n)
            n.extend([len(n), len(n)])
            return _MultisetHistogram(n)
        m = dict(zip(s, range(len(s))))
        d = dict(zip(range(len(s)), [0]*len(s)))
        for i in n:
            d[m[i]] += 1
        return _multiset_histogram(d)


def nP(n, k=None, replacement=False):
    """Return the number of permutations of ``n`` items taken ``k`` at a time.

    Possible values for ``n``::
        integer - set of length ``n``
        sequence - converted to a multiset internally
        multiset - {element: multiplicity}

    If ``k`` is None then the total of all permutations of length 0
    through the number of items represented by ``n`` will be returned.

    If ``replacement`` is True then a given item can appear more than once
    in the ``k`` items. (For example, for 'ab' permutations of 2 would
    include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in
    ``n`` is ignored when ``replacement`` is True but the total number
    of elements is considered since no element can appear more times than
    the number of elements in ``n``.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import nP
    >>> from sympy.utilities.iterables import multiset_permutations, multiset
    >>> nP(3, 2)
    6
    >>> nP('abc', 2) == nP(multiset('abc'), 2) == 6
    True
    >>> nP('aab', 2)
    3
    >>> nP([1, 2, 2], 2)
    3
    >>> [nP(3, i) for i in range(4)]
    [1, 3, 6, 6]
    >>> nP(3) == sum(_)
    True

    When ``replacement`` is True, each item can have multiplicity
    equal to the length represented by ``n``:

    >>> nP('aabc', replacement=True)
    121
    >>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)]
    [1, 3, 9, 27, 81]
    >>> sum(_)
    121

    References
    ==========

    http://en.wikipedia.org/wiki/Permutation

    See Also
    ========
    sympy.utilities.iterables.multiset_permutations

    """
    try:
        n = as_int(n)
    except ValueError:
        return Integer(_nP(_multiset_histogram(n), k, replacement))
    return Integer(_nP(n, k, replacement))


@cacheit
def _nP(n, k=None, replacement=False):
    from sympy.functions.combinatorial.factorials import factorial
    from sympy.core.mul import prod

    if k == 0:
        return 1
    if isinstance(n, SYMPY_INTS):  # n different items
        # assert n >= 0
        if k is None:
            return sum(_nP(n, i, replacement) for i in range(n + 1))
        elif replacement:
            return n**k
        elif k > n:
            return 0
        elif k == n:
            return factorial(k)
        elif k == 1:
            return n
        else:
            # assert k >= 0
            return _product(n - k + 1, n)
    elif isinstance(n, _MultisetHistogram):
        if k is None:
            return sum(_nP(n, i, replacement) for i in range(n[_N] + 1))
        elif replacement:
            return n[_ITEMS]**k
        elif k == n[_N]:
            return factorial(k)/prod([factorial(i) for i in n[_M] if i > 1])
        elif k > n[_N]:
            return 0
        elif k == 1:
            return n[_ITEMS]
        else:
            # assert k >= 0
            tot = 0
            n = list(n)
            for i in range(len(n[_M])):
                if not n[i]:
                    continue
                n[_N] -= 1
                if n[i] == 1:
                    n[i] = 0
                    n[_ITEMS] -= 1
                    tot += _nP(_MultisetHistogram(n), k - 1)
                    n[_ITEMS] += 1
                    n[i] = 1
                else:
                    n[i] -= 1
                    tot += _nP(_MultisetHistogram(n), k - 1)
                    n[i] += 1
                n[_N] += 1
            return tot


@cacheit
def _AOP_product(n):
    """for n = (m1, m2, .., mk) return the coefficients of the polynomial,
    prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients
    of the product of AOPs (all-one polynomials) or order given in n.  The
    resulting coefficient corresponding to x**r is the number of r-length
    combinations of sum(n) elements with multiplicities given in n.
    The coefficients are given as a default dictionary (so if a query is made
    for a key that is not present, 0 will be returned).

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import _AOP_product
    >>> from sympy.abc import x
    >>> n = (2, 2, 3)  # e.g. aabbccc
    >>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand()
    >>> c = _AOP_product(n); dict(c)
    {0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1}
    >>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)]
    True

    The generating poly used here is the same as that listed in
    http://tinyurl.com/cep849r, but in a refactored form.

    """
    from collections import defaultdict

    n = list(n)
    ord = sum(n)
    need = (ord + 2)//2
    rv = [1]*(n.pop() + 1)
    rv.extend([0]*(need - len(rv)))
    rv = rv[:need]
    while n:
        ni = n.pop()
        N = ni + 1
        was = rv[:]
        for i in range(1, min(N, len(rv))):
            rv[i] += rv[i - 1]
        for i in range(N, need):
            rv[i] += rv[i - 1] - was[i - N]
    rev = list(reversed(rv))
    if ord % 2:
        rv = rv + rev
    else:
        rv[-1:] = rev
    d = defaultdict(int)
    for i in range(len(rv)):
        d[i] = rv[i]
    return d


def nC(n, k=None, replacement=False):
    """Return the number of combinations of ``n`` items taken ``k`` at a time.

    Possible values for ``n``::
        integer - set of length ``n``
        sequence - converted to a multiset internally
        multiset - {element: multiplicity}

    If ``k`` is None then the total of all combinations of length 0
    through the number of items represented in ``n`` will be returned.

    If ``replacement`` is True then a given item can appear more than once
    in the ``k`` items. (For example, for 'ab' sets of 2 would include 'aa',
    'ab', and 'bb'.) The multiplicity of elements in ``n`` is ignored when
    ``replacement`` is True but the total number of elements is considered
    since no element can appear more times than the number of elements in
    ``n``.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import nC
    >>> from sympy.utilities.iterables import multiset_combinations
    >>> nC(3, 2)
    3
    >>> nC('abc', 2)
    3
    >>> nC('aab', 2)
    2

    When ``replacement`` is True, each item can have multiplicity
    equal to the length represented by ``n``:

    >>> nC('aabc', replacement=True)
    35
    >>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)]
    [1, 3, 6, 10, 15]
    >>> sum(_)
    35

    If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k``
    then the total of all combinations of length 0 hrough ``k`` is the
    product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity
    of each item is 1 (i.e., k unique items) then there are 2**k
    combinations. For example, if there are 4 unique items, the total number
    of combinations is 16:

    >>> sum(nC(4, i) for i in range(5))
    16

    References
    ==========

    * http://en.wikipedia.org/wiki/Combination
    * http://tinyurl.com/cep849r

    See Also
    ========
    sympy.utilities.iterables.multiset_combinations
    """
    from sympy.functions.combinatorial.factorials import binomial
    from sympy.core.mul import prod

    if isinstance(n, SYMPY_INTS):
        if k is None:
            if not replacement:
                return 2**n
            return sum(nC(n, i, replacement) for i in range(n + 1))
        assert k >= 0
        if replacement:
            return binomial(n + k - 1, k)
        return binomial(n, k)
    if isinstance(n, _MultisetHistogram):
        N = n[_N]
        if k is None:
            if not replacement:
                return prod(m + 1 for m in n[_M])
            return sum(nC(n, i, replacement) for i in range(N + 1))
        elif replacement:
            return nC(n[_ITEMS], k, replacement)
        # assert k >= 0
        elif k in (1, N - 1):
            return n[_ITEMS]
        elif k in (0, N):
            return 1
        return _AOP_product(tuple(n[_M]))[k]
    else:
        return nC(_multiset_histogram(n), k, replacement)


@cacheit
def _stirling1(n, k):
    if n == k == 0:
        return S.One
    if 0 in (n, k):
        return S.Zero
    n1 = n - 1

    # some special values
    if n == k:
        return S.One
    elif k == 1:
        return factorial(n1)
    elif k == n1:
        return C.binomial(n, 2)
    elif k == n - 2:
        return (3*n - 1)*C.binomial(n, 3)/4
    elif k == n - 3:
        return C.binomial(n, 2)*C.binomial(n, 4)

    # general recurrence
    return n1*_stirling1(n1, k) + _stirling1(n1, k - 1)


@cacheit
def _stirling2(n, k):
    if n == k == 0:
        return S.One
    if 0 in (n, k):
        return S.Zero
    n1 = n - 1

    # some special values
    if k == n1:
        return C.binomial(n, 2)
    elif k == 2:
        return 2**n1 - 1

    # general recurrence
    return k*_stirling2(n1, k) + _stirling2(n1, k - 1)


def stirling(n, k, d=None, kind=2, signed=False):
    """Return Stirling number S(n, k) of the first or second (default) kind.

    The sum of all Stirling numbers of the second kind for k = 1
    through n is bell(n). The recurrence relationship for these numbers
    is::

    {0}       {n}   {0}      {n + 1}     {n}   {  n  }
    { } = 1;  { } = { } = 0; {     } = j*{ } + {     }
    {0}       {0}   {k}      {  k  }     {k}   {k - 1}

    where ``j`` is::
        ``n`` for Stirling numbers of the first kind
        ``-n`` for signed Stirling numbers of the first kind
        ``k`` for Stirling numbers of the second kind

    The first kind of Stirling number counts the number of permutations of
    ``n`` distinct items that have ``k`` cycles; the second kind counts the
    ways in which ``n`` distinct items can be partitioned into ``k`` parts.
    If ``d`` is given, the "reduced Stirling number of the second kind" is
    returned: ``S^{d}(n, k) = S(n - d + 1, k - d + 1)`` with ``n >= k >= d``.
    (This counts the ways to partition ``n`` consecutive integers into
    ``k`` groups with no pairwise difference less than ``d``. See example
    below.)

    To obtain the signed Stirling numbers of the first kind, use keyword
    ``signed=True``. Using this keyword automatically sets ``kind`` to 1.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import stirling, bell
    >>> from sympy.combinatorics import Permutation
    >>> from sympy.utilities.iterables import multiset_partitions, permutations

    First kind (unsigned by default):

    >>> [stirling(6, i, kind=1) for i in range(7)]
    [0, 120, 274, 225, 85, 15, 1]
    >>> perms = list(permutations(range(4)))
    >>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)]
    [0, 6, 11, 6, 1]
    >>> [stirling(4, i, kind=1) for i in range(5)]
    [0, 6, 11, 6, 1]

    First kind (signed):

    >>> [stirling(4, i, signed=True) for i in range(5)]
    [0, -6, 11, -6, 1]

    Second kind:

    >>> [stirling(10, i) for i in range(12)]
    [0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0]
    >>> sum(_) == bell(10)
    True
    >>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2)
    True

    Reduced second kind:

    >>> from sympy import subsets, oo
    >>> def delta(p):
    ...    if len(p) == 1:
    ...        return oo
    ...    return min(abs(i[0] - i[1]) for i in subsets(p, 2))
    >>> parts = multiset_partitions(range(5), 3)
    >>> d = 2
    >>> sum(1 for p in parts if all(delta(i) >= d for i in p))
    7
    >>> stirling(5, 3, 2)
    7

    References
    ==========

    * http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
    * http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

    See Also
    ========
    sympy.utilities.iterables.multiset_partitions

    """
    # TODO: make this a class like bell()

    n = as_int(n)
    k = as_int(k)
    if n < 0:
        raise ValueError('n must be nonnegative')
    if k > n:
        return S.Zero
    if d:
        # assert k >= d
        # kind is ignored -- only kind=2 is supported
        return _stirling2(n - d + 1, k - d + 1)
    elif signed:
        # kind is ignored -- only kind=1 is supported
        return (-1)**(n - k)*_stirling1(n, k)

    if kind == 1:
        return _stirling1(n, k)
    elif kind == 2:
        return _stirling2(n, k)
    else:
        raise ValueError('kind must be 1 or 2, not %s' % k)


@cacheit
def _nT(n, k):
    """Return the partitions of ``n`` items into ``k`` parts. This
    is used by ``nT`` for the case when ``n`` is an integer."""
    if k == 0:
        return 1 if k == n else 0
    return sum(_nT(n - k, j) for j in range(min(k, n - k) + 1))


def nT(n, k=None):
    """Return the number of ``k``-sized partitions of ``n`` items.

    Possible values for ``n``::
        integer - ``n`` identical items
        sequence - converted to a multiset internally
        multiset - {element: multiplicity}

    Note: the convention for ``nT`` is different than that of ``nC`` and``nP`` in that
    here an integer indicates ``n`` *identical* items instead of a set of
    length ``n``; this is in keepng with the ``partitions`` function which
    treats its integer-``n`` input like a list of ``n`` 1s. One can use
    ``range(n)`` for ``n`` to indicate ``n`` distinct items.

    If ``k`` is None then the total number of ways to partition the elements
    represented in ``n`` will be returned.

    Examples
    ========

    >>> from sympy.functions.combinatorial.numbers import nT

    Partitions of the given multiset:

    >>> [nT('aabbc', i) for i in range(1, 7)]
    [1, 8, 11, 5, 1, 0]
    >>> nT('aabbc') == sum(_)
    True

    (TODO The following can be activated with >>> when
    taocp_multiset_permutation is in place.)
    >> [nT("mississippi", i) for i in range(1, 12)]
    [1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1]

    Partitions when all items are identical:

    >>> [nT(5, i) for i in range(1, 6)]
    [1, 2, 2, 1, 1]
    >>> nT('1'*5) == sum(_)
    True

    When all items are different:

    >>> [nT(range(5), i) for i in range(1, 6)]
    [1, 15, 25, 10, 1]
    >>> nT(range(5)) == sum(_)
    True

    References
    ==========

    * http://undergraduate.csse.uwa.edu.au/units/CITS7209/partition.pdf

    See Also
    ========
    sympy.utilities.iterables.partitions
    sympy.utilities.iterables.multiset_partitions

    """
    from sympy.utilities.iterables import multiset_partitions

    if isinstance(n, SYMPY_INTS):
        # assert n >= 0
        # all the same
        if k is None:
            return sum(_nT(n, k) for k in range(1, n + 1))
        return _nT(n, k)
    if not isinstance(n, _MultisetHistogram):
        try:
            # if n contains hashable items there is some
            # quick handling that can be done
            u = len(set(n))
            if u == 1:
                return nT(len(n), k)
            elif u == len(n):
                n = range(u)
            raise TypeError
        except TypeError:
            n = _multiset_histogram(n)
    N = n[_N]
    if k is None and N == 1:
        return 1
    if k in (1, N):
        return 1
    if k == 2 or N == 2 and k is None:
        m, r = divmod(N, 2)
        rv = sum(nC(n, i) for i in range(1, m + 1))
        if not r:
            rv -= nC(n, m)//2
        if k is None:
            rv += 1  # for k == 1
        return rv
    if N == n[_ITEMS]:
        # all distinct
        if k is None:
            return bell(N)
        return stirling(N, k)
    if k is None:
        return sum(nT(n, k) for k in range(1, N + 1))
    tot = 0
    for p in multiset_partitions(
            [i for i, j in enumerate(n[_M]) for ii in range(j)]):
        tot += len(p) == k
    return tot

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