A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911735804329984 |
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| author | Dong, Yao Xu, Chao |
| author_facet | Dong, Yao Xu, Chao |
| contents | Motivated by a recent conjecture of Bsila, Cox, Hugo, Styron and Zhuang concerning fixed points and pixed points on pattern-avoiding permutations, we prove a bivariate refinement involving descent statistics. Given a set of permutations $Π$, let $\mathfrak{S}_n(Π)$ denote the set of permutations in the symmetric group $\mathfrak{S}_n$ that avoid every element of $Π$ in the sense of pattern avoidance. For each set $Π$ appearing in their conjecture, we show that the pairs of statistics $(\mathrm{des},\mathrm{fix})$ and $(\mathrm{ides},\mathrm{pix})$ are equidistributed over $\mathfrak{S}_n(Π)$. Our proof is based on explicit ordinary generating functions for the corresponding pattern-avoiding classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_00646 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations Dong, Yao Xu, Chao Combinatorics 05A05, 05A15, 05A19 Motivated by a recent conjecture of Bsila, Cox, Hugo, Styron and Zhuang concerning fixed points and pixed points on pattern-avoiding permutations, we prove a bivariate refinement involving descent statistics. Given a set of permutations $Π$, let $\mathfrak{S}_n(Π)$ denote the set of permutations in the symmetric group $\mathfrak{S}_n$ that avoid every element of $Π$ in the sense of pattern avoidance. For each set $Π$ appearing in their conjecture, we show that the pairs of statistics $(\mathrm{des},\mathrm{fix})$ and $(\mathrm{ides},\mathrm{pix})$ are equidistributed over $\mathfrak{S}_n(Π)$. Our proof is based on explicit ordinary generating functions for the corresponding pattern-avoiding classes. |
| title | A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations |
| topic | Combinatorics 05A05, 05A15, 05A19 |
| url | https://arxiv.org/abs/2606.00646 |