﻿ 2021解析组合与计数组合系列报告会

# 2021解析组合与计数组合系列报告会

摘要： A sequence $P_n(x)$ of polynomials in $x$ is holonomic (P-recursive) if it satisfies a linear recurrence with polynomial coefficients in $x$ and $n$. Many polynomial sequences from combinatorics, representation theory and number theory are shown to be holonomic. It is natural and fundamental to study the degree pattern of holonomic polynomial sequences. We will present a classification of the degree growth of such sequences and explain two applications related to combinatorial identities and exponential sums over finite fields respectively. This is a joint work with Jason P. Bell, Daqing Wan, Rong-Hua Wang and Hang Yin.

摘要： A finite or infinite matrix with integer or real coefficients is called totally positive if all its minors are nonnegative. Such matrices have a wide variety of applications across pure and applied mathematics. In this talk, we exhibit a low-triangular matrix of polynomials in six indeterminates that appears empirically to be coefficientwise totally positive. We prove the coefficientwise total positivity of a special case of such matrices, which includes the reversed Stirling subset triangle.

摘要：  Stieltjes moment sequences and Hamburger moment sequences play an important role in different mathematical branches. In this talk, we will report some results for moment properties of combinatorial sequences.

摘要：  I will present some intriguing connections between pattern avoiding permutations and lattice walks.

摘要：  Denote by $\mathfrak{S}_{M,i}$  the set of multipermutations, in which the element in the first position is fixed as an integer $i$, on a multiset $M=\{1^{p_1},\ldots,n^{p_n}\}$. Let $A_{{M},i}(x,q)$ be the joint distribution polynomial of descents and major index of multipermutations in $\mathfrak{S}_{M,i}$. In this talk, we will first discuss a calculation formula for $A_{{M},i}(x,q)$.  $({M},i)$-multiset Eulerian polynomials $A_{M,i}(x)$ are the descent polynomials of multipermutations in $\mathfrak{S}_{M,i}$. We will show that $xA_{M,i}(x)$, $c_1A_{M,i}(x)+c_2A_{M,j}(x)$ and $c_1xA_{M,i}(x)+c_2A_{M,j}(x)$ have only real roots, where $c_1$ and $c_2$ are nonnegative real number, $i,j\in M$ and $i<j$. Use $[n]_k$ to denote the the multiset $\{1^k,2^k,\ldots,n^k\}$.  It is also shown that $A_{[n]_k,i}(x)$ is reciprocal with $A_{[n]_k,n-i+1}(x)$ for any $1\leq i\leq n$. We also prove that $A_{[n]_k,i}(x)+A_{[n]_k,n-i+1}(x)$ and $xA_{[n]_k,i}(x)+A_{[n]_k,n-i+1}(x)$ are $\gamma$-positive, and $A_{[n]_k,i}(x)$ is bi-$\gamma$-positive. Taking $k=1$, write $A_{[n]_k,i}(x)=A_{n,i}(x)$ for short. We give a combinatorial interpretation for $\gamma$-coefficients of $A_{n,i}(x)+A_{n,n-i+1}(x)$ for any $1\leq i\leq n$.

摘要：  In this talk, we discuss the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems. This generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including up-down run polynomials for symmetric groups, alternating run polynomials for hyperoctahedral groups, flag descent polynomials for hyperoctahedral groups and flag ascent-plateau polynomials for Stirling permutations.

摘要：  The concept of the edge cover polynomial was first introduced by Akbari and Oboudi. It was shown to have some interesting properties and connect with many other well known polynomials associated with graphs. Noticing it is often of interest to study the average graph polynomials, we intended to observe the edge cover polynomials from this angle. Our stimulation also partly comes from some independent work on various kinds of average graph polynomials, especially the work of Brown et al. on average independence polynomials. Specifically, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph K_n, based on which we conclude that the average edge cover polynomial of order n is unimodal and has at least n-3 non-real roots.

摘要：  The noncrossing partitions and their relatives have been extensively studied. In this talk,  we introduce a new family  of noncrossing partitions named as [k]-conflicting noncrossing partitions,  based upon our bijection between the ballot paths and the closed flows on forks. We will present the structural and enumerative properties of the [k]-conflicting noncrossing partitions.

摘要：  We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we give a common generalization of eigenvalue inequalities for (Hermitian) normalized Laplacian matrices of simple (signed, weighted, directed) graphs. Besides, it is well known that the eigenvalues of totally positive matrices are all real. We give a unified proof of the interlacing properties of eigenvalues of principle submatrices of totally positive matrices.