Avoiding a pair of patterns in multisets and compositions

TL;DR

This paper classifies Wilf-type equivalences among pattern pairs in multiset permutations and compositions, providing a comprehensive understanding of their structural similarities and differences.

Contribution

It identifies all multiset and composition pattern equivalences for certain pattern lengths using diverse combinatorial techniques, extending to infinite families.

Findings

Complete classification of (3,3) and (3,4) pattern pairs in compositions.

Identification of all equivalences among length-3 and length-4 pattern pairs.

Extension of results to infinite families of pattern pairs.

Abstract

In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To establish our results, we make use of a variety of techniques, including Ferrers-equivalence arguments, sorting by minimal/maximal letters, analysis of active sites and direct bijections. In several cases, our arguments may be extended to prove multiset equivalences for infinite families of pattern pairs. Our results apply equally well to the Wilf-type classification of compositions, and as a consequence, we obtain a complete description of the Wilf-equivalence classes for pairs of patterns of type (3,3) and (3,4) on compositions, with the possible exception of two classes of type (3,4).