Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers

A D-permutation is a permutation of $[2n]$ satisfying $2k-1 \le σ(2k-1)$ and $2k \ge σ(2k)$ for all $k$; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.

Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions

We introduce the generic Lah polynomials $L_{n,k}(ϕ)$, which enumerate unordered forests of increasing ordered trees with a weight $ϕ_i$ for each vertex with $i$ children. We show that, if the weight sequence $ϕ$ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials $L_n(ϕ,y)$ is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive typ…

A class of multivariate polynomial convolutions (and applications)

We prove two "master" convolution theorems for multivariate determinantal polynomials. The methods used include basic properties of what we call a "minor-orthogonal" ensemble as well as properties of the mixed discriminant of matrices. We also give applications, including a rederivation of a result of Barvinok on computing the permanent of a low rank matrix and a polynomial convolution corresponding to the unitarily invariant addition of generalized singular values.

使用 Foata--Zeilberger 双射及其变体的 Laguerre 有向图解释的连分数

Continued fractions using a Laguerre digraph interpretation of the Foata--Zeilberger bijection and its variants

本文用 Laguerre 有向图解释 Foata--Zeilberger 双射以计算排列中的环，从而得到了枚举排列和 D-排列的多变量多项式的 Jacobi 型、Thron 型和 Stieltjes 型连分数，并证明了 Sokal、Zeng 等人提出的连分数猜想。

循环交替排列的连分数

Continued fractions for cycle-alternating permutations

该研究针对循环交替排列的计数多元多项式得到Stieltjes型连分数，引入其推广结构交替Laguerre有向图并给出对应计数多项式的指数生成函数，还从图的角度解释了相关连分数对应的两类Rogers矩阵。

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity

We define an infinite sequence of generalizations, parametrized by an integer $m \ge 1$, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for $m$-Dyck and $m$-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of …