A Catalan Subset of Descending Plane Partitions

TL;DR

This paper identifies a Catalan-numbered subset within descending plane partitions and constructs a generating tree structure similar to 231-avoiding permutations, aiming to aid in finding bijections with other combinatorial objects.

Contribution

It isolates a Catalan subset of descending plane partitions and constructs a generating tree analogous to 231-avoiding permutations, providing new structural insights.

Findings

Identified a Catalan subset within descending plane partitions.

Constructed a generating tree matching the structure of 231-avoiding permutations.

Provides a framework for potential bijections with other Catalan-structured objects.

Abstract

Descending plane partitions, alternating sign matrices, and totally symmetric self-complementary plane partitions are equinumerous combinatorial sets for which no explicit bijection is known. In this paper, we isolate a subset of descending plane partitions counted by the Catalan numbers. The proof follows by constructing a generating tree on these descending plane partitions that has the same structure as the generating tree for 231-avoiding permutations. We hope this result will provide insight on the search for a bijection with alternating sign matrices and/or totally symmetric self-complementary plane partitions, since these also contain Catalan subsets.