Simple marked mesh patterns

TL;DR

This paper systematically studies the distributions of simple marked mesh patterns in permutations, providing explicit generating functions, recursions, and discussing q-analogues, with modifications and enumerative results.

Contribution

It introduces the first systematic analysis of simple marked mesh patterns, including explicit formulas, recursions, and modifications, advancing permutation pattern research.

Findings

Explicit generating functions for simple marked mesh patterns

Recursion formulas for counting pattern occurrences

Enumerative results for modified mesh patterns

Abstract

In this paper we begin the first systematic study of distributions of simple marked mesh patterns. Mesh patterns were introduced recently by Br\"and\'en and Claesson in connection with permutation statistics. We provide explicit generating functions in several general cases, and develop recursions to compute the numbers in question in some other cases. Certain $q$-analogues are discussed. Moreover, we consider two modifications of the notion of a marked mesh pattern and provide enumerative results for them.