Interlacing log-concavity of the derangement polynomials and the Eulerian polynomials

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• 作者：Cindy C.Y. Gu^a ; ^{cindygutjut@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Larry X.W. Wang^b ; ^{wsw82@nankai.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：58
• 期：Complete
• 页码：52-60
• 全文大小：436 K

文摘

Let $D(n,k)$ be the set of derangements of [n]$\left[n\right]$ with k$k$ excedances and d(n,k)$d(n,k)$ be the cardinality of $D(n,k)$. We establish a bijection between $D(n,k)$ and the set of labeled lattice paths of length n$n$ with k$k$ horizontal edges. Using this bijection, we give a direct combinatorial proof of the inequalities d(n,k−1)d(m,l+1)<d(n,k)d(m,l)$d(n,k-1)d(m,l+1)<d(n,k)d(m,l)$ for n≥m≥1$n\ge m\ge 1$ and l≥k≥1$l\ge k\ge 1$. Moreover, we prove the interlacing log-concavity of the sequences {d(n,k)}_0≤k≤n${\left\{d(n,k)\right\}}_{0\le k\le n}$. By a similar combinatorial structure, we show that the Eulerian polynomials possess these properties.