论文标题
几乎是fibibonacci避免模式排列的统计数据
Statistics on Almost-Fibonacci Pattern-Avoiding Permutations
论文作者
论文摘要
我们证明$ | av_n（231,312,1432）| $，$ | av_n（312,321,1342）| $ $ $ | av_n（231,312,4321,21543）| $ $，$ | $，$ |其中$ f_n $是$ n $ -th fibonacci使用惯例$ f_0 = f_1 = 1 $和$ av_n（s）$是所有长度$ n $排列的集合，避免了集合$ s $中的所有模式。为此，我们根据斐波那契排列来表征这些集合中排列的结构。然后，我们使用统计数据（例如反转数和统计量）进一步量化结构，该统计量衡量了斐波那契子序的长度。最后，我们用斐波那契排列的生成函数编写的函数编码这些统计信息。我们使用这些生成功能来找到有关斐波那契身份的复发关系和加法公式的类似物。
We prove that $|Av_n(231,312,1432)|$, $|Av_n(312,321,1342)|$ $|Av_n(231,312,4321,21543)|$, and $ |Av_n(321,231,4123,21534)|$, are all equal to $F_{n+1} - 1$ where $F_n$ is the $n$-th Fibonacci number using the convention $F_0 = F_1 = 1$ and $Av_n(S)$ is the set of all permutations of length $n$ that avoid all of the patterns in the set $S$. To do this, we characterize the structures of the permutations in these sets in terms of Fibonacci permutations. Then, we further quantify the structures using statistics such as inversion number and a statistic that measures the length of Fibonacci subsequences. Finally, we encode these statistics in generating functions written in terms of the generating function for Fibonacci permutations. We use these generating functions to find analogs about recurrence relation and addition formulae of Fibonacci identities.
