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Jiang Zeng

Publications and source records attributed to Jiang Zeng.

At least 73 records · Page 4Linked to original sources
arXiv2012

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.

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arXiv2012

A generalization of the Mehta-Wang determinant and Askey-Wilson polynomials

Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $\det((a+j-i)Γ(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $\Pf((j-i)Γ(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.

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arXiv2012

On Zudilin's q-question about Schmidt's problem

We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740] derived from q-Pfaff-Saalschutz identity.

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arXiv2012

Jacobi-Stirling polynomials and $P$-partitions

We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind $ \JS(n+k,n;z)$ by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting $\JS(n+k,n;z)=p_{k,0}(n)+p_{k,1}(n)z+...+p_{k,k}(n)z^k$, we show that $(1-t)^{3k-i+1}\sum_{n\geq0}p_{k,i}(n)t^n$ is a polynomial in $t$ with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley's theory of $P$-partitions.

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arXiv2012

New congruences for sums involving Apery numbers or central Delannoy numbers

The Apéry numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\sum_{k=0}^{n}{n+k\choose 2k}^2{2k\choose k}^2, \quad D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences: \sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\equiv \sum_{k=0}^{n-1}\varepsilon^k (2k+1)^{2r+1}D_k \equiv 0\pmod n, where $n\geqslant 1$, $r\geqslant 0$, and $\varepsilon=\pm1$. For $r=1$, we further show that \sum_{k=0}^{n-1}(2k+1)^{3}A_k &\equiv 0\pmod{n^3}, \quad \sum_{k=0}^{p-1}(2k+1)^{3}A_k &\equiv p^3 \pmod{2p^6}, where $p>3$ is a prime. The following congruence \sum_{k=0}^{n-1} {n+k\choose k}^2{n-1\choose k}^2 \equiv 0 \pmod{n} plays an important role in our proof.

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arXiv2012

A symmetrical q-Eulerian identity

We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq 0}\binom{a+b}{k}A_{k,a-1}=\sum_{k\geq 0}\binom{a+b}{k}A_{k,b-1}. {equation*} We give two proofs, using generating function and bijections, respectively.

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arXiv2012

Two new triangles of $q$-integers via $q$-Eulerian polynomials of type $A$ and $B$

The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogue of this expansion for Carlitz's $q$-Eulerian polynomials as well as a similar formula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall give some applications of these two formulae, which involve two new sequences of polynomials in the variable $q$ with positive integral coefficients. An open problem is to give a combinatorial interpretation for these polynomials.

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arXiv2012

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Brändén (European J. Combin. \textbf{29} (2008), no.~2, 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 (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that of Josuat-Vergès (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).

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arXiv2011

On the number of congruence classes of paths

Let $P_n$ denote the undirected path of length $n-1$. The cardinality of the set of congruence classes induced by the graph homomorphisms from $P_n$ onto $P_k$ is determined. This settles an open problem of Michels and Knauer (Disc. Math., 309\ (2009)\ 5352-5359). Our result is based on a new proven formula of the number of homomorphisms between paths.

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arXiv2011

An explicit formula for the linearization coefficients of Bessel polynomials

We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat's formulas in their proof of the positivity results about these coefficients (Constructive Approximation, 27(2008), 15-32). As a bonus we also obtain a generalization of an integral formula of Boros and Moll (J. Comput. Appl. Math. 106 (1999), 361-368).

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arXiv2011

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 $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}ε^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$ \sum_{k=0}^{n}ε^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{2n\choose n}, $$ where $ε=\pm 1$.

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arXiv2010

Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants

Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions.

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arXiv2010

A q-analogue of Catalan Hankel determinants

In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schröder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.

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arXiv2010

Bijections for Entringer families

André proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of André's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of $E_{n,k}$ in the model of trees. The aim of this paper is to provide bijections between the different models for $E_{n,k}$ as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's alternating permutation model and Poupard's increasing tree model.

math.CO↗
arXiv2010

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $λ= 1^{e_1}2^{e_2} \ldots$ of a tree on the vertex set $\set{1,2,\ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a $q$-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.

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arXiv2010

Orthogonal polynomials and operator orderings

An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.

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