论文标题
避开图案的鱼潮排列和上升序列
Pattern-Avoiding Fishburn Permutations and Ascent Sequences
论文作者
论文摘要
Fishburn置换是避免双感模式$（231，\ {1 \}，\ {1 \}）$的排列，而上升序列是一系列非阴性整数的序列，其中每个条目都比其左侧的数量高于或等于一个。 Fishburn排列和上升序列与Bookext $ g $的Bousquet-Mélou，Claesson，Dukes和Kitaev相连。 We write $F_n(σ_1,\ldots,σ_k)$ to denote the set of Fishburn permutations of length $n$ which avoid each of $σ_1,\ldots,σ_k$ and we write $A_n(α_1,\ldots,α_k)$ to denote the set of ascent sequences which avoid each of $α_1，\ ldots，α_k$。我们通过表明$ g $限制了$ f_n（3412）$和$ a_n（201）$之间的bloodive限制，我们解决了GIL和Weiner的猜想。在GIL和Weiner的工作基础上，我们使用基本技术来枚举$ f_n（123）$相对于反转数量和从左到右的最大值的数量，以$ q $ $ binmial系数获得表达式，并枚举所有$ f_n（123，σ）$ $ f_n（123，σ）$。我们使用生成的树技术来研究$ f_n（321，1423）$，$ f_n（321,3124）$和$ f_n（321,2143）$的生成功能，相对于反转数量和左至右Maxima的数量。我们使用这些结果显示$ | f_n（321,1423）| = | f_n（321,3124）| = f_ {n+2} - n -1 $，其中$ f_n $是fibonacci编号，$ | f_n（321,2143）| = 2^{n-1} $。我们以各种猜想和开放性问题结束。
A Fishburn permutation is a permutation which avoids the bivincular pattern $(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of ascents to its left. Fishburn permutations and ascent sequences are linked by a bijection $g$ of Bousquet-Mélou, Claesson, Dukes, and Kitaev. We write $F_n(σ_1,\ldots,σ_k)$ to denote the set of Fishburn permutations of length $n$ which avoid each of $σ_1,\ldots,σ_k$ and we write $A_n(α_1,\ldots,α_k)$ to denote the set of ascent sequences which avoid each of $α_1,\ldots,α_k$. We settle a conjecture of Gil and Weiner by showing that $g$ restricts to a bijection between $F_n(3412)$ and $A_n(201)$. Building on work of Gil and Weiner, we use elementary techniques to enumerate $F_n(123)$ with respect to inversion number and number of left-to-right maxima, obtaining expressions in terms of $q$-binomial coefficients, and to enumerate $F_n(123,σ)$ for all $σ$. We use generating tree techniques to study the generating functions for $F_n(321, 1423)$, $F_n(321,3124)$, and $F_n(321,2143)$ with respect to inversion number and number of left-to-right maxima. We use these results to show $|F_n(321,1423)| = |F_n(321,3124)| = F_{n+2} - n - 1$, where $F_n$ is a Fibonacci number, and $|F_n(321,2143)| = 2^{n-1}$. We conclude with a variety of conjectures and open problems.
