Permutation totally symmetric self-complementary plane partitions

TL;DR

This paper establishes a bijection between a subset of symmetric plane partitions and permutations, introduces new partial orders on these objects, and connects them to well-known lattices like Tamari and Catalan.

Contribution

It identifies a permutation subset within symmetric plane partitions, defines new partial orders, and links these to classical lattices and Bruhat order, advancing combinatorial understanding.

Findings

Bijection between certain plane partitions and permutations.

New partial order on permutations containing Tamari and Catalan lattices.

Distributive lattice structure related to Bruhat order.

Abstract

Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify a subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new partial order on totally symmetric self-complementary plane partitions arising from this perspective and show that this is a distributive lattice related to Bruhat order when restricted to permutations.