An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in recent years. This paper is an attempt at finding a closed- form formula for the number of S-restricted compositions of n. To do so, we reduce the problem to finding solutions to corresponding so-called interpreters which are linear homogeneous recurrence relations with constant coefficients. Then, we reduce interpreters to Diophantine equations. Such equations are not in general solvable. Thus, we restrict our attention to those S-restricted composition problems whose interpreters have a small number of coefficients, thereby leading to solvable Diophantine equations. The formalism developed is then used to study the integer sequences related to some well-known cases of the S-restricted composition problem.
Another approach to the problem of @math -restricted compositions is to reduce it to some other known problems. For example, it has been shown that some cases of the problem are in bijective correspondence with some classes of restricted binary strings and pattern-avoiding permutations @cite_19 . It has also been shown that the number of @math -restricted compositions equals specific Fibonacci numbers for some specific sets @math @cite_43 .
@cite_19 @cite_43
2182636777 2076917081 2096790857 2146489966
We study several restricted classes of compositions by giving one-to-one maps between them and different classes of restricted binary strings or pattern avoiding permutations. Inspired by the ECO method [8], new succession rules for these classes are presented. Finally, we obtain generating algorithms in Constant Amortized Time (CAT) for theses classes. We show that among the compositions of n into positive parts, the number g(n) that avoid a given pattern @p of three letters is independent of @p. We find the generating function of g(n) , and it shows that the sequence g(n) is not P-recursive. If S is a given multiset, we show that the number of permutations of S that avoid a pattern @p of three letters is independent of @p. Finally, we give a bijective proof of the fact that if M=1^a^"^1...k^a^"^k is a given multiset then the number of permutations of M that avoid the pattern (123) is a symmetric function of the multiplicities a"1,...,a"k. The bijection uses the Greene-Kleitman symmetric chain decomposition of the Boolean lattice. We define the notion of @math -free for locally restricted compositions, which means roughly that if such a composition contains a part @math and nearby parts are at least @math smaller, then @math can be replaced by any larger part. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part and number distinct parts, all accurate to @math . We study compositions of the integer @math in which the first part, successive differences, and the last part are constrained to lie in prescribed sets @math , respectively. A simple condition on @math guarantees that the generating function @math has only a simple pole on its circle of convergence and this at @math , a function independent of @math and @math . Thus the number of compositions is asymptotic to @math for a suitable constant @math . We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of @math , such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If @math has only finitely many positive elements @math , or finitely many negative elements @math , then the largest part and number of distinct part sizes are almost surely @math . On the other hand, when both @math and @math have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely @math , and we give sufficient conditions for the largest part to be almost surely asymptotic to @math .