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, 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.
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BibTeX
@misc{biswas2026matchings,
title = {{Matchings and shape-Wilf-Equivalence of sets of patterns of length three I: Triples}},
author = {Sucharita Biswas and Umesh Shankar and Sivaramakrishnan Sivasubramanian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.08562},
eprint = {2609.08562},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08562},
}
RIS
TY - UNPB
TI - Matchings and shape-Wilf-Equivalence of sets of patterns of length three I: Triples
AU - Sucharita Biswas
AU - Umesh Shankar
AU - Sivaramakrishnan Sivasubramanian
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.08562
UR - https://arxiv.org/abs/2609.08562
AB - 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, 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.
ER -