论文标题
置换模式字符多项式的积极性
Positivity of permutation pattern character polynomials
论文作者
论文摘要
令$n_σ（π）$表示s_n $中排列$π\ in s_k $ in s_k $中的置换模式的出现数量。 Gaetz and Ryba（2021）表明使用分区代数表明，$ d $ them $ m_ {σ，d，n}（π）（π）$ $n_σ$ $n_σ$ $n_σ$ $π$由$ n，m_1，m_1，m_ de $ n $ n $ n $ n $ n $ n $ y $ mm_i的$ n_i y $ y $ m m_i i $ mm_i i $ im y i $ m m_i， $π$。他们还表明，系数$ \langleχ^{λ[n]}，m_ {σ，d，d，n} \ rangle $符合$ n $ in $ n $的多项式$ a_ {σ，d}^λ（n）$。这项工作是由猜想的激励，即当$σ= \ text {id} _k $是身份排列时，所有这些系数都是无负的。我们直接计算多项式的封闭式$ a _ {\ text {id} _k}^λ（n）$在情况下$λ=（1），（1,1），$和$（2）$，并使用它来验证这些案例的积极性来证明这些案例与polynomials相比，与所有$ k $ k $ k $ k $ k相比，我们还研究了$a_σ^{（1）}（n）$的情况，为此，我们为多项式及其领先系数提供了一个公式。
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.
