Fibonacci numbers, consecutive patterns, and inverse peaks

TL;DR

This paper explores the enumeration of permutations avoiding certain patterns, revealing connections to Fibonacci numbers and providing multiple proofs using diverse combinatorial techniques.

Contribution

It introduces new formulas for permutation enumeration based on inverse peak statistics and demonstrates their relation to Fibonacci sequences, with elementary consequences.

Findings

Enumeration formulas involve Fibonacci-related sequences

Unique permutation with zero inverse peak in fixed descent classes

Multiple proof methods including generating functions and tilings

Abstract

We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases is given by a sequence related to Fibonacci numbers. We also show that there is exactly one permutation whose inverse peak number is zero among all permutations with any fixed descent composition, and we give a few elementary consequences of this fact. Our proofs involve generating functions, symmetric functions, regular expressions, and monomino-domino tilings.