On (joint) equidistributions of mesh patterns 123 and 132 with symmetric shadings

TL;DR

This paper investigates the joint equidistributions of mesh patterns 123 and 132 with symmetric shadings, establishing 75 bijective proofs of their equidistribution and exploring the maximum number of such pairs.

Contribution

It provides the first systematic bijective proofs of joint equidistributions of mesh patterns 123 and 132 with symmetric shadings, extending prior work on pattern distributions.

Findings

Established 75 joint equidistributions of mesh patterns 123 and 132.

Proved 36 non-symmetric equidistributions.

Identified a maximum of 93 potential symmetric equidistributed pairs.

Abstract

A notable problem within permutation patterns that has attracted considerable attention in literature since 1973 is the search for a bijective proof demonstrating that 123-avoiding and 132-avoiding permutations are equinumerous, both counted by the Catalan numbers. Despite this equivalence, the distributions of occurrences of the patterns 123 and 132 are distinct. When considering 123 and 132 as mesh patterns and selectively shading boxes, similar scenarios arise, even when avoidance is defined by the Bell numbers or other sequences, rather than the Catalan numbers. However, computer experiments suggest that mesh patterns 123 and 132 may indeed be equidistributed. Furthermore, by considering symmetric shadings relative to the anti-diagonal, a maximum of 93 such equidistributed pairs can potentially exist. This paper establishes 75 such equidistributions, leaving the justification of the remaining cases as open problems. As a by-product, we also prove 36 relevant non-symmetric equidistributions. All our proofs are bijective and involve swapping occurrences of the patterns in question, thereby demonstrating their joint equidistribution. Our findings are a continuation of the systematic study of distributions of short-length mesh patterns initiated by Kitaev and Zhang in 2019.