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Eulerian-type polynomials over matchings and matching permutations

Shi-Mei Ma, Sergey Kitaev, Jean Yeh, Yeong-Nan Yeh

TL;DR

This work builds a deep link between matchings and permutations by aligning quadruple permutation statistics with matching statistics via new polynomial frameworks. It introduces $(p,q)$-Eulerian polynomials and $(s,t)$-even-odd larger matching polynomials, proving a symmetry and a direct expression in terms of classical Eulerian polynomials, and provides an exponential generating function that encodes these relations. A grammar-based proof establishes the main identity connecting matchings to permutations, while extensions to $q$-derangement and type B Eulerian polynomials reveal rich specialization behavior. The third part develops matching permutations and neighbor-enriched polynomials, proving $e$-positivity for the NCA and NCR families and linking them to trivariate second-order Eulerian polynomials via explicit substitutions. Overall, the paper unifies several strands of combinatorics on matchings, permutations, and Eulerian-type statistics, generalizing prior results and enabling new symmetric and positivity results.

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.

Eulerian-type polynomials over matchings and matching permutations

TL;DR

This work builds a deep link between matchings and permutations by aligning quadruple permutation statistics with matching statistics via new polynomial frameworks. It introduces -Eulerian polynomials and -even-odd larger matching polynomials, proving a symmetry and a direct expression in terms of classical Eulerian polynomials, and provides an exponential generating function that encodes these relations. A grammar-based proof establishes the main identity connecting matchings to permutations, while extensions to -derangement and type B Eulerian polynomials reveal rich specialization behavior. The third part develops matching permutations and neighbor-enriched polynomials, proving -positivity for the NCA and NCR families and linking them to trivariate second-order Eulerian polynomials via explicit substitutions. Overall, the paper unifies several strands of combinatorics on matchings, permutations, and Eulerian-type statistics, generalizing prior results and enabling new symmetric and positivity results.

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.
Paper Structure (9 sections, 20 theorems, 82 equations, 3 figures)

This paper contains 9 sections, 20 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $(p,q)$-Eulerian polynomials and $(s,t)$-even-odd larger matching polynomials
  3. Definitions and main results
  4. Proof of Theorem \ref{['mainthm01']}
  5. $q$-derangement polynomials and $(p,q)$-Eulerian polynomials of type $B$
  6. Matching permutations, neighbor polynomials and $e$-positivity
  7. Definitions and preliminaries
  8. Main results
  9. Relationship to trivariate second-order Eulerian polynomials

Key Result

Theorem 1

Let $\operatorname{lne}$ (resp. $\operatorname{lcr}$, $\operatorname{nal}$, $\operatorname{lrp}$) be the left-nesting (resp. left-crossing, neighbor alignment, LR pair) statistic, where a LR pair is a pair of consecutive integer $(i,i+1)$ in the arc diagram of $M\in\mathcal{M}_n$ such that $i$ is an

Figures (3)

  • Figure 1: Blocks in each matching are ordered in increasing order of their second elements.
  • Figure 2: The correspondence between $\mathcal{M}_2$ and $\operatorname{MP}_2$.
  • Figure 3: The 0-1-2 increasing plane trees on $[3]$.

Theorems & Definitions (28)

  • Theorem 1
  • Example 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Proposition 6: Dumont96
  • Lemma 7: Ma24
  • Lemma 8
  • proof
  • proof
  • ...and 18 more