More bijective Catalan combinatorics on permutations and on signed permutations

TL;DR

This paper introduces bijections linking Dyck paths, noncrossing partitions, and 231-avoiding permutations, preserving key statistics and extending to signed permutations, advancing combinatorial understanding.

Contribution

It constructs new bijections that preserve important combinatorial statistics and generalizes these to signed permutations, enriching Catalan combinatorics.

Findings

Bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations.

Preservation of area, inversion number, and major index statistics.

Extensions of these bijections to signed permutations.

Abstract

In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.