Matchings and shape-Wilf-Equivalence of sets of patterns of length three I: Triples

Abstract: Permutation pattern avoidance on Ferrers boards has become a central topic in enumerative combinatorics with important connections to matchings, set partitions, and other combinatorial structures as it allows one to build families of Wilf-equivalent patterns. While shape-Wilf-equivalence classes have been completely determined for individual patterns and pairs of patterns of length three, the corresponding classification for larger pattern sets has remained open. In this paper, we provide a complete classification of the shape-Wilf-equivalence classes of triples of patterns of length three. Our proofs use a bijective encoding of pattern avoiding transversals to establish all equivalence classes. As an application, we enumerate matchings avoiding triples of patterns of length three for all but two equivalence classes, extending previous results of Bloom and Elizalde. These enumerative results identify additional families of combinatorial objects counted by the Fuss-Catalan numbers and by other integer sequences appearing in the OEIS.