Bounded Littlewood identity related to alternating sign matrices

TL;DR

This paper establishes a bounded version of a Littlewood identity related to alternating sign matrices and provides combinatorial interpretations, aiming to deepen understanding of their connections to plane partitions.

Contribution

It introduces a bounded generalization of a Littlewood identity and offers combinatorial interpretations for both sides, advancing the theoretical framework.

Findings

Established a bounded version of the generalized identity.

Provided combinatorial interpretations for both sides of the identity.

Aims to facilitate a combinatorial proof connecting alternating sign trapezoids and plane partitions.

Abstract

An identity that is reminiscent of the Littlewood identity plays a fundamental role in recent proofs of the facts that alternating sign triangles are equinumerous with totally symmetric self-complementary plane partitions and that alternating sign trapezoids are equinumerous with holey cyclically symmetric lozenge tilings of a hexagon. We establish a bounded version of a generalization of this identity. Further, we provide combinatorial interpretations of both sides of the identity. The ultimate goal would be to construct a combinatorial proof of this identity (possibly via an appropriate variant of the Robinson-Schensted-Knuth correspondence) and its unbounded version as this would improve the understanding of the relation between alternating sign trapezoids and plane partition objects.