Canon Permutation Posets

TL;DR

This paper studies canon permutations, a special class of permutations with a uniform subsequence structure, revealing their descent polynomials have a product form, are palindromic, and connecting them to Stanley's $(P, au)$-partitions.

Contribution

It generalizes the concept of canon permutations using poset theory, providing new proofs, extending results, and exploring connections to $ ext{(P,} au)$-partitions, including $ ext{γ}$-positivity and dissonant canon permutations.

Findings

Descent polynomial of canon permutations has a product structure.

When $P$ is graded, descent polynomials are palindromic.

Introduction of dissonant canon permutations and $ ext{γ}$-positivity results.

Abstract

A permutation of the multiset $\{1^m,2^m,\dots,n^m\}$ is a {\em canon permutation} if the subsequence formed by the $j$th copy of each element of $[n]:=\{1,2,\dots,n\}$ is identical for all $j\in[m]$. Canon permutations were introduced by Elizalde and are motivated by pattern-avoiding concepts such as (quasi-)Stirling permutations. He proved that the descent polynomial of canon permutations exhibits a surprising product structure; as a further consequence, it is palindromic. Our goal is to understand canon permutations from the viewpoint of Stanley's $(P,\omega)$-partitions, along the way generalizing Elizalde's definition and results. We start with a labeled poset $P$ and extend it in a natural way to canon labelings of the product poset $P \times [n]$. The resulting descent polynomial has a product structure which arises naturally from the theory of $(P,\omega)$-partitions and simplifies existing proofs. When $P$ is graded, this theory also implies palindromicity. We include results on weak descent polynomials, an amphibian construction between canon permutations and multiset permutations, giving rise to \emph{dissonant canon permutations}, as well as $\gamma$-positivity and interpretations of descent polynomials of canon permutations.