ABSTRACTGiven a permutation , we say an index i is a peak if πi − 1 πi + 1. Let P(π) denote the set of peaks of π. Given any set S of positive integers, define . Billey–Burdzy–Sagan showed that for all fixed subsets of positive integers S and sufficiently large n, for some polynomial pS(x) depending on S. They conjectured that the coefficients of pS(x) expanded in a binomial coefficient basis centered at max (S) are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of pS(x). Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at 0, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.
Another new result in @cite_7 shows that the number of permutations with the same peak set for signed permutations can be enumerated using the peak polynomial @math for unsigned permutations. We present an alternate proof that can be used to reduce many signed permutation statistic problems to unsigned permutation statistic problems. We denote the group of signed permutations as @math .
@cite_7
2240013452 2076917081 1691157804 2145801920
A signed permutation = in the hyperoctahedral group B_n is a word such that each -n, , -1, 1, , n and | |, | |, , | | = 1,2, ,n . An index i is a peak of if i-1 i+1 and P_B( ) denotes the set of all peaks of . Given any set S, we define P_B(S,n) to be the set of signed permutations B_n with P_B( ) = S. In this paper we are interested in the cardinality of the set P_B(S,n). In 2012, Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, S_n. In this paper we extend their results to the hyperoctahedral group; in particular we show that #P_B(S,n) = p(n)2^ 2n-|S|-1 where p(n) is the same polynomial found in by Billey, Burdzy and Sagan which leads to the explicit computation of interesting special cases of the polynomial p(n). In addition we have extended these results to the case where we add =0 at the beginning of the permutations, which gives rise to the possibility of a peak at position 1, for both the symmetric and the hyperoctahedral groups. 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. Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be the length of the longest increasing subsequence. For example, if N = 5 and 7r is the permutation 5 1 3 2 4 (in one-line notation: thus 7r(1) = 5, 7r(2) = 1, ... ), then the longest increasing subsequences are 1 2 4 and 1 3 4, and N() = 3. Equip SN with uniform distribution, A shuffle consists of a permutation and re-encryption of a set of input ciphertexts. One application of shuffles is to build mix-nets. We suggest an honest verifier zero-knowledge argument for the correctness of a shuffle of homomorphic encryptions. Our scheme is more efficient than previous schemes both in terms of communication and computation. The honest verifier zero-knowledge argument has a size that is independent of the actual cryptosystem being used and will typically be smaller than the size of the shuffle itself. Moreover, our scheme is well suited for the use of multi-exponentiation and batch-verification techniques. Additionally, we suggest a more efficient honest verifier zero-knowledge argument for a commitment containing a permutation of a set of publicly known messages. We also suggest an honest verifier zero-knowledge argument for the correctness of a combined shuffle-and-decrypt operation that can be used in connection with decrypting mix-nets based on ElGamal encryption. All our honest verifier zero-knowledge arguments can be turned into honest verifier zero-knowledge proofs. We use homomorphic commitments as an essential part of our schemes. When the commitment scheme is statistically hiding we obtain statistical honest verifier zero-knowledge arguments; when the commitment scheme is statistically binding, we obtain computational honest verifier zero-knowledge proofs.