Pattern avoidance in matchings and partitions

TL;DR

This paper extends pattern avoidance concepts to matchings and partitions, providing enumeration, algebraic generating functions, bijections, and classifications that deepen understanding of combinatorial pattern structures.

Contribution

It introduces new pattern avoidance classes in matchings and partitions, derives algebraic generating functions, and classifies pattern pairs based on shape-Wilf-equivalence.

Findings

Enumerated 312-avoiding matchings and partitions with algebraic generating functions

Provided a direct proof of Bóna's formula for 1342-avoiding permutations

Classified pairs of length-3 patterns by shape-Wilf-equivalence

Abstract

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of B\'ona for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies existing proofs by Backelin--West--Xin and Jel\'{\i}nek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.