Canon Permutation Posets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915411893682176 |
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| author | Beck, Matthias Deligeorgaki, Danai |
| author_facet | Beck, Matthias Deligeorgaki, Danai |
| contents | 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 $(P,ω)$-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 $P \times [n]$. The resulting descent polynomial has a product structure which arises naturally from the theory of $(P,ω)$-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 $γ$-positivity and interpretations of descent polynomials of canon permutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canon Permutation Posets Beck, Matthias Deligeorgaki, Danai Combinatorics 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 $(P,ω)$-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 $P \times [n]$. The resulting descent polynomial has a product structure which arises naturally from the theory of $(P,ω)$-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 $γ$-positivity and interpretations of descent polynomials of canon permutations. |
| title | Canon Permutation Posets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.03245 |