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

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 jointly equidistributed. Furthermore, by considering symmetric shadings relative to the diagonal, a maximum of 93 equidistributed pairs can potentially exist. This paper establishes 75 joint equidistributions, leaving the justification of the remaining cases as open problems. As a by-product, we also prove 36 relevant non-symmetric joint 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.