Determinantal formulae for a symmetric generating function of totally symmetric plane partitions

TL;DR

This paper introduces determinantal formulae for a symmetric generating function of totally symmetric plane partitions, connecting it to lattice path models, tableaux, and generalizations of Littlewood identities.

Contribution

It presents new determinantal formulae and lattice path models for symmetric plane partition polynomials, expanding their combinatorial and algebraic understanding.

Findings

Derived determinantal formulae for the generating function.

Established connections to lattice path models and tableaux.

Linked the polynomials to generalizations of dual Littlewood identities.

Abstract

Ilse Fischer and the second author introduced in [Algebr. Comb. 7 (2024), no. 5, 1319-1345] a two parameter family of polynomials defined as sums over totally symmetric plane partitions and connected to alternating sign matrices and cyclically symmetric lozenge tilings of a hexagon with a triangular hole. In this paper we present several determinantal formulae leading to new lattice path models and a novel family of tableaux. The later illustrates that the polynomials of our interest can be thought of as generalisations of the three dual Littlewood identities.