On Permutation Weights and $q$-Eulerian Polynomials

TL;DR

This paper studies permutation weights and $q$-Eulerian polynomials, proving coefficient stabilization, deriving recurrence relations, and exploring combinatorial properties of related power series.

Contribution

It proves the stabilization of coefficients in $q$-Eulerian polynomials and introduces a recurrence relation, advancing understanding of permutation weights and their combinatorial significance.

Findings

Coefficients of $E_n(x, q)$ stabilize as $n$ increases.

A recurrence relation for $E_n(x, q)$ is established.

A recursive formula for certain integer partitions is derived.

Abstract

Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingr\'imsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a permutation $\sigma$ viewed as a sequence of integers, computing the weight of $\sigma$ involves recursively counting descents of certain subpermutations of $\sigma$. Using this weight function, one can define a $q$-analog $E_n(x,q)$ of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials $E_n(x,q)$. First, we show that the coefficients of $E_n(x, q)$ stabilize as $n$ goes to infinity, which was conjectured by Dugan, Glennon, Gunnells, and Steingr\'imsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019), and enables the definition of the formal power series $W_d(t)$, which has interesting combinatorial properties. Second, we derive a recurrence relation for $E_n(x, q)$, similar to the known recurrence for the classical Eulerian polynomials $A_n(x)$. Finally, we give a recursive formula for the numbers of certain integer partitions and, from this, conjecture a recursive formula for the stabilized coefficients mentioned above.