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Restricted inversion polynomials

Jeongwon Lee, Nathan Lesnevich, Martha Precup

TL;DR

The paper generalizes descent polynomials to restricted inversion polynomials $\mathcal{I}_{\mathbf{h}}(S;n)$ by counting $\mathbf{h}$-inversions in $S_n$. It proves polynomiality in $n$ for $n\ge \mathbf{j}(S)$, and provides two explicit binomial-expansions with nonnegative coefficients, along with a graded and a $q$-analogue version. It establishes log-concavity for the coefficient sequences in both expansions via height polynomials and poset methods, and discusses a $q$-analogue with a corresponding expansion and conjectures about strong $q$-log-concavity. The results extend descent-polynomial theory, connect to Hessenberg-variety combinatorics, and raise natural questions about graded log-concavity in the $q$-setting.

Abstract

For a finite subset $I$ of positive integers, the descent polynomial $\mathcal{D}(I;n)$ counts the number of permutations in $S_n$ that have descent set $I$. We generalize descent polynomials by considering permutations with a specific subset $S$ of common inversions called $\mathbf{h}$-inversions, where $\mathbf{h} = (\mathbf{h}(1), \mathbf{h}(2), \ldots )$ is a weakly increasing sequence of positive integers such that $\mathbf{h}(i)> i$. We prove that this more general count, denoted by $\mathcal{I}_\mathbf{h}(S;n)$, is also a polynomial. We give three explicit expansions for $\mathcal{I}_\mathbf{h}(S;n)$, prove the coefficients for two of these expansions are log-concave, and define a graded generalization.

Restricted inversion polynomials

TL;DR

The paper generalizes descent polynomials to restricted inversion polynomials by counting -inversions in . It proves polynomiality in for , and provides two explicit binomial-expansions with nonnegative coefficients, along with a graded and a -analogue version. It establishes log-concavity for the coefficient sequences in both expansions via height polynomials and poset methods, and discusses a -analogue with a corresponding expansion and conjectures about strong -log-concavity. The results extend descent-polynomial theory, connect to Hessenberg-variety combinatorics, and raise natural questions about graded log-concavity in the -setting.

Abstract

For a finite subset of positive integers, the descent polynomial counts the number of permutations in that have descent set . We generalize descent polynomials by considering permutations with a specific subset of common inversions called -inversions, where is a weakly increasing sequence of positive integers such that . We prove that this more general count, denoted by , is also a polynomial. We give three explicit expansions for , prove the coefficients for two of these expansions are log-concave, and define a graded generalization.

Paper Structure

This paper contains 12 sections, 21 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Inversion polynomials
  3. Restricted inversion sets
  4. Polynomality
  5. Two expansion formulas
  6. Expansion Formula 1: The bk(S)-expansion
  7. Expansion Formula 2: The ak(S)-expansion
  8. Log-concavity
  9. Height polynomials
  10. Log-concavity for bk(S)
  11. Log-concavity for ak(S)
  12. A q-analogue of the h-inversion polynomial

Key Result

Theorem 1.1

Let $\mathbf{h} = (\mathbf{h}(1), \mathbf{h}(2), \ldots)$ be a weakly increasing sequence of positive integers such that $\mathbf{h}(i)>i$ and $S\subset \{(i,j) \mid i<j\leq \mathbf{h}(i)\}$. Let $\mathbf{j}(S) := \max\{j \mid (i,j) \in S\}$. Then $\mathcal{I}_\mathbf{h}(S;n)$ is given by a polynomi

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 53 more