Séminaire Lotharingien de Combinatoire, 93B.60 (2025), 11 pp.
Matthias Beck and Danai Deligeorgaki
Canon Permutation Posets
Abstract.
A permutation of the multiset {1m,2m,...,nm}$ is a canon permutation if
the subsequence formed by the jth copy of each element of [n]:={1,2,...,n} is identical for
all j ∈ [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,ω)-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 × [n]. The resulting descent
polynomial has a product structure which arises naturally from the theory of (P,ω)-partitions. 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, as well as
γ-positivity and interpretations of descent polynomials of canon permutations.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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