Eulerian-type polynomials over matchings and matching permutations

TL;DR

This paper explores the deep connection between matchings and permutations, revealing new symmetric properties and polynomial relationships that generalize previous combinatorial results.

Contribution

It introduces matching permutations, establishes a symmetric expansion of neighbor polynomials, and proves e-positivity of NCA-polynomials, extending known Eulerian polynomial relationships.

Findings

Matchings with no left-nestings correspond to permutation statistics.

Symmetric distribution of certain matching statistics is proven.

Relationship between neighbor polynomials and second-order Eulerian polynomials is established.

Abstract

Claesson and Linusson [Proc. Am. Math. Soc., 139 (2011), 435-449] observed that there are n! matchings on [2n] with no left-nestings. Inspired by this result, this paper is devoted to exploring a deeper connection between matchings and permutations. We first discover that a quadruple statistic over matchings corresponds to the well known quadruple statistic (exc,drop,fix,cyc) over permutations, where exc, drop, fix and cyc are the excedance, drop, fixed point and cycle statistics, respectively. By introducing matching permutations, we provide a symmetric expansion of a five-variable neighbor polynomial of matchings, which encodes a great deal of neighbor information. As an application, we discover the e-positivity of NCA-polynomials, which implies that the left-nesting number, the left-crossing number and the neighbor alignment number are distributed symmetrically over all matchings on [2n]. We also establish the relationship between the five-variable neighbor polynomials and the trivariate second-order Eulerian polynomials, which generalizes the related results of Claesson and Linusson, Cameron and Killpatrick as well as Chen and Fu.