Harmonic numbers, Catalan's triangle and mesh patterns

TL;DR

This paper explores the combinatorial properties of small mesh patterns, linking pattern avoidance to harmonic numbers and Catalan triangle distributions, and introduces new sequences related to Catalan numbers.

Contribution

It establishes connections between mesh pattern avoidance and harmonic numbers, Catalan triangle distributions, and introduces new sequences counted by Catalan numbers.

Findings

Avoidance of one mesh pattern linked to harmonic numbers

Distribution of three mesh patterns on 132-avoiding permutations given by Catalan triangle

Identification of Wilf-equivalence between two mesh patterns

Abstract

The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we link avoidance of one of the patterns to the harmonic numbers, while for three other patterns we show their distributions on 132-avoiding permutations are given by the Catalan triangle. Also, we show that two specific mesh patterns are Wilf-equivalent. As a byproduct of our studies, we define a new set of sequences counted by the Catalan numbers and provide a relation on the Catalan triangle that seems to be new.