Jiang Zeng

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Branden [P. Branden, Actions on permutations and unimodality of descent polynomials, European J. Combin. 29 (2) (2008) 514-531] about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a (p,q)-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The (p,q)-analogue unifies and generalizes our recent results [H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin. 31 (7) (2010) 1689-1705] and that of Josuat-Verges [M. Josuat-Verges, A q-enumeration of alternating permutations, European J. Combin. 31 (7) (2010) 1892-1906].

Factors of alternating sums of products of binomial and q-binomial coefficients

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack n_1}^{-1}\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\choose 2}} \prod_{i=1}^m {n_i+n_{i+1}\brack n_i+k}\in \N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\brack n}^{-1}{2j\brack j} \sum_{k=j}^n(-1)^{n-k}q^{A}\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\brack n-k}{k+j\brack k-j}^r\in N[q], where A=(r-1){n\choose 2}+r{j+1\choose 2}+{k\choose 2}-rjk, which solves a problem raised by Zudilin [Electron. J. Combin. 11 (2004), #R22].

Some q-analogues of supercongruences of Rodriguez-Villegas

Abstract We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence ∑ k = 0 p − 1 ( 2 k k ) 2 16 k ≡ ( − 1 p ) ( mod p 2 ) , where ( ⋅ p ) is the Legendre symbol, has the following two nice q-analogues: ∑ k = 0 p − 1 ( q ; q 2 ) k 2 ( q 2 ; q 2 ) k 2 q ( 1 + e ) k ≡ ( − 1 p ) q ( p 2 − 1 ) e 4 ( mod ( 1 + q + ⋯ + q p − 1 ) 2 ) , where ( a ; q ) n = ( 1 − a ) ( 1 − a q ) ⋯ ( 1 − a q n − 1 ) and e = ± 1 . Several related conjectures are also proposed.

Factors of binomial sums from the Catalan triangle

Abstract By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n 1 , … , n m , n m + 1 = n 1 , and any nonnegative integer r, the expression n 1 − 1 ( n 1 + n m n 1 ) − 1 ∑ k = 1 n 1 k 2 r + 1 ∏ i = 1 m ( n i + n i + 1 n i + k ) is either an integer or a half-integer. Moreover, several related conjectures are proposed.

Two truncated identities of Gauss

Two new expansions for partial sums of Gauss@? triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function p@?(n) and for the partition function pod(n) counting the partitions of n with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.

Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials

Abstract The Apery polynomials are defined by A n ( x ) = ∑ k = 0 n ( n k ) 2 ( n + k k ) 2 x k for all nonnegative integers n. We confirm several conjectures of Z.-W. Sun on the congruences for the sum ∑ k = 0 n − 1 ( − 1 ) k ( 2 k + 1 ) A k ( x ) with x ∈ Z .

Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems. The separation of variables technique naturally leads to integral representations of combinatorial numbers where the integrand contains a product of one or more types of orthogonal polynomials. This also establishes the positivity of such integrals.

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set {1,2,...,n}, the local direction of each edge (ij) is from i to j if i

Symmetric unimodal expansions of excedances in colored permutations

We consider several generalizations of the classical γ -positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the γ -positivity for Eulerian polynomials for derangements of type B . More general expansion formulae are also given for Eulerian polynomials for r -colored derangements. Our results answer and generalize several recent open problems in the literature.

FACTORS OF SUMS AND ALTERNATING SUMS INVOLVING BINOMIAL COEFFICIENTS AND POWERS OF INTEGERS

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n1,…,nm, nm+1 = n1, and any nonnegative integer r, there holds \[ \sum_{k=0}^{n_1}\varepsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} \left(\begin{array}{@{}c@{}} n_i+n_{i+1}+1\\ n_i-k \end{array}\right) \equiv 0\quad\!\left(\!{\rm mod}\ (n_1+n_m+1)\left(\begin{array}{@{}c@{}} n_1+n_m\\ n_1\end{array}\right)\right)\!, \] and conjecture that for any nonnegative integer r and positive integer s such that r + s is odd,