Dyck Paths, Standard Young Tableaux, and Pattern Avoiding Permutations

TL;DR

This paper develops a generating function and counting formulas for permutations avoiding or containing specific patterns, linking these to Dyck paths and Young tableaux, and applies the results to optimize a card game strategy.

Contribution

It introduces a new enumeration formula for pattern-avoiding permutations and establishes bijections with Dyck paths and Young tableaux, extending previous statistical analyses.

Findings

Derived a two-variable generating function for pattern-avoiding permutations

Established a bijection between permutations, Dyck paths, and Young tableaux

Applied enumeration results to optimize a card game strategy

Abstract

We present a generating function and a closed counting formula in two variables that enumerate a family of classes of permutations that avoid or contain an increasing pattern of length three and have a prescribed number of occurrences of another pattern of length three. This gives a refinement of some previously studied statistics, most notably one by Noonan. The formula is also shown to enumerate a family of classes of Dyck paths and Standard Young Tableaux, and a bijection is given between the corresponding classes of these two families of objects. Finally, the results obtained are used to solve an optimization problem for a certain card game.