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| Format: | Preprint |
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2022
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| Accès en ligne: | https://arxiv.org/abs/2204.10633 |
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| _version_ | 1866910378708959232 |
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| author | Gaetz, Christian Pierson, Laura |
| author_facet | Gaetz, Christian Pierson, Laura |
| contents | Let $N_σ(π)$ denote the number of occurrences of a permutation pattern $σ\in S_k$ in a permutation $π\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{σ,d,n}(π)$ of $N_σ$ on the conjugacy class of $π$ is given by a polynomial in $n,m_1,\dots,m_{dk}$, where $m_i$ denotes the number of $i$-cycles of $π$. They also showed that the coefficient $\langle χ^{λ[n]}, M_{σ,d,n}\rangle$ agrees with a polynomial $a_{σ,d}^λ(n)$ in $n$. This work is motivated by the conjecture that when $σ=\text{id}_k$ is the identity permutation, all of these coefficients are nonnegative. We directly compute closed forms for the polynomials $a_{\text{id}_k}^λ(n)$ in the cases $λ=(1),(1,1),$ and $(2)$, and use this to verify the positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than $k$. We also study the case $a_σ^{(1)}(n)$, for which we give a formula for the polynomials and their leading coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_10633 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Positivity of permutation pattern character polynomials Gaetz, Christian Pierson, Laura Combinatorics Let $N_σ(π)$ denote the number of occurrences of a permutation pattern $σ\in S_k$ in a permutation $π\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{σ,d,n}(π)$ of $N_σ$ on the conjugacy class of $π$ is given by a polynomial in $n,m_1,\dots,m_{dk}$, where $m_i$ denotes the number of $i$-cycles of $π$. They also showed that the coefficient $\langle χ^{λ[n]}, M_{σ,d,n}\rangle$ agrees with a polynomial $a_{σ,d}^λ(n)$ in $n$. This work is motivated by the conjecture that when $σ=\text{id}_k$ is the identity permutation, all of these coefficients are nonnegative. We directly compute closed forms for the polynomials $a_{\text{id}_k}^λ(n)$ in the cases $λ=(1),(1,1),$ and $(2)$, and use this to verify the positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than $k$. We also study the case $a_σ^{(1)}(n)$, for which we give a formula for the polynomials and their leading coefficients. |
| title | Positivity of permutation pattern character polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2204.10633 |