Statistics on Almost-Fibonacci Pattern-Avoiding Permutations

TL;DR

This paper establishes that certain pattern-avoiding permutations have sizes related to Fibonacci numbers, characterizes their structures, and explores their statistics through generating functions connected to Fibonacci identities.

Contribution

It characterizes the structure of specific pattern-avoiding permutations in terms of Fibonacci permutations and derives related generating functions and identities.

Findings

Sizes of permutation sets equal Fibonacci numbers minus one

Structural characterization of these permutations in terms of Fibonacci permutations

Derived generating functions and Fibonacci identities for these permutations

Abstract

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.