对合上的欧拉分布确实是$γ$ -正的
The Eulerian distribution on involutions is indeed $γ$-positive
摘要：
令$\mathcal I_n$和$\mathcal J_n$分别表示$\{1, \dots, n\}$的对合和无定点对合的集合，并且令$\text{des}(π)$表示置换$π$的下降数。我们证明了郭和曾的一个猜想，即$I_n(t) := \sum_{π\in \mathcal I_n} t^{\text{des}(π)}$对$n \ge 1$是$γ$阳性的，而$J_{2n}(t) := \sum_{π\in \mathcal J_{2n}} t^{\text{des}(π)}$对$n \ge 9$是$γ$阳性的.我们还证明了具有$m$双下降和$k$下降的$(3412, 3421)$ -避免置换的个数等于具有$m$双下降和$k$下降的可分置换的个数.
英文摘要：
Let $\mathcal I_n$ and $\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\{1, \dots, n\}$, respectively, and let $\text{des}(π)$ denote the number of descents of the permutation $π$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \sum_{π\in \mathcal I_n} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 1$ and $J_{2n}(t) := \sum_{π\in \mathcal J_{2n}} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable permutations with $m$ double descents and $k$ descents.
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