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限制置换、斐波那契数和k-广义斐波那契数

Restricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci Numbers

Eric S. Egge, Toufik Mansour

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29 pages
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摘要：

当$π$不包含具有与$σ$相同的所有成对比较的子序列时，置换$π\in S_n$被称为{ \it避免}置换$σ\in S_k$。对于任何排列集合$R$，我们写$S_n(R)$来表示$S_n$中避免$R$中的每个排列的排列集合。1985年，Simion和Schmidt证明了$|S_n(132, 213, 123)|$等于斐波那契数$F_{n+1}$。在本文中，我们推广这个结果在几个方面。本文首先利用Mansour的一个结果证明了对于某个无穷置换族中的任意置换$τ$，$|S_n(132, 213, τ)|$都是用斐波那契数或$k$ -广义斐波那契数表示的.在许多情况下，我们给出明确的枚举，我们证明双射。然后，我们使用生成函数技术表明，对于任何排列$γ$在第二个无限家庭的排列，$|S_n(123, 132, γ)|$也给出了斐波那契数或$k$ -广义斐波那契数。在许多情况下，我们给出明确的枚举，其中一些我们证明双射。我们继续使用生成函数的技术表明，对于任何排列$ω$在第三无限家庭的排列，$|S_n(132, 2341, ω)|$给出的斐波那契数，并为任何排列$μ$在第四无限家庭的排列，$|S_n(132, 3241, μ)|$给出的斐波那契数和$k$ -广义斐波那契数。在某些情况下，我们给出了明确的枚举。最后，我们给出了一个无限类的例子，一组$R$的排列，其中$|S_n(R)|$满足一个线性齐次常系数递归关系。

英文摘要：

A permutation $π\in S_n$ is said to {\it avoid} a permutation $σ\in S_k$ whenever $π$ contains no subsequence with all of the same pairwise comparisons as $σ$. For any set $R$ of permutations, we write $S_n(R)$ to denote the set of permutations in $S_n$ which avoid every permutation in $R$. In 1985 Simion and Schmidt showed that $|S_n(132, 213, 123)|$ is equal to the Fibonacci number $F_{n+1}$. In this paper we generalize this result in several ways. We first use a result of Mansour to show that for any permutation $τ$ in a certain infinite family of permutations, $|S_n(132, 213, τ)|$ is given in terms of Fibonacci numbers or $k$-generalized Fibonacci numbers. In many cases we give explicit enumerations, which we prove bijectively. We then use generating function techniques to show that for any permutation $γ$ in a second infinite family of permutations, $|S_n(123, 132, γ)|$ is also given in terms of Fibonacci numbers or $k$-generalized Fibonacci numbers. In many cases we give explicit enumerations, some of which we prove bijectively. We go on to use generating function techniques to show that for any permutation $ω$ in a third infinite family of permutations, $|S_n(132, 2341, ω)|$ is given in terms of Fibonacci numbers, and for any permutation $μ$ in a fourth infinite family of permutations, $|S_n(132, 3241, μ)|$ is given in terms of Fibonacci numbers and $k$-generalized Fibonacci numbers. In several cases we give explicit enumerations. We conclude by giving an infinite class of examples of a set $R$ of permutations for which $|S_n(R)|$ satisfies a linear homogeneous recurrence relation with constant coefficients.

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