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Abstract:A permutation of the multiset $\{1^m,2^m,\dots,n^m\}$ is a 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 dissonant canon permutations, as well as $\gamma$-positivity and interpretations of descent polynomials of canon permutations.

Submission history

From: Danai Kalliopi Deligeorgaki [view email]
[v1] Fri, 4 Oct 2024 09:10:52 UTC (22 KB)
[v2] Sun, 27 Jul 2025 04:19:16 UTC (24 KB)
[v3] Sun, 13 Sep 2026 18:22:24 UTC (24 KB)