Enumeration of bilaterally symmetric 3-noncrossing partitions

TL;DR

This paper develops a new enumeration technique for bilaterally symmetric 3-noncrossing partitions, extending existing combinatorial correspondences and providing a Maple package for related lattice walk problems.

Contribution

It introduces a novel approach for counting bilaterally symmetric 3-noncrossing partitions using a new technique and a Maple package for vacillating lattice walk enumeration.

Findings

Developed a Maple package for 2D vacillating lattice walk enumeration.

Derived relations for special bilaterally symmetric partitions.

Extended combinatorial correspondences to 3-noncrossing partitions.

Abstract

Schutzenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3-noncrossing partitions, we use a different technique to develop a Maple package for 2-dimensional vacillating lattice walk enumeration problems. The package also applies to the hesitating case. As applications, we find several interesting relations for some special bilaterally symmetric partitions.