Matchings and shape-Wilf-Equivalence of sets of patterns of length three II: Quadruples and Quintuples

Abstract

Building on our classification of shape-Wilf-equivalence classes for triples of patterns of length three, we complete the classification for quadruples and quintuples. The larger pattern sets exhibit structural features that are not captured by the encoding methods used for triples and require additional combinatorial tools. Our main new ingredient is a Dyck-path approach to Ferrers boards containing the staircase board. By introducing distinguished cells and related structural parameters on the associated Dyck paths, we obtain new shape-Wilf-equivalences and, for several classes, explicit formulas for the number of avoiding transversals on a fixed Ferrers board. We further study the corresponding pattern-avoiding perfect matchings through the pattern-preserving bijection of Bloom and Elizalde. Besides deriving recurrence relations, we give bijective interpretations of several matching enumeration sequences in terms of grand Dyck paths, Schröder paths, and Dyck paths. These constructions provide additional combinatorial structure beyond the classification itself. Together with the companion paper on triples, the present work completes the classification of shape-Wilf-equivalence classes for subsets of patterns of length three, apart from the trivial empty and full pattern sets.