Inversions and the Gog-Magog problem

TL;DR

This paper explores the combinatorial relationship between Gog and Magog triangles, introduces new trapezoid structures, and conjectures their equienumeration, providing explicit bijections and analyzing inversion distributions.

Contribution

It introduces left Gog and GOGAm trapezoids, conjectures their equienumeration, and offers explicit bijections for certain cases, advancing understanding of these combinatorial objects.

Findings

Conjecture that Gog and GOGAm trapezoids are equienumerated.

Explicit bijections for trapezoids with one or two diagonals.

Analysis of inversion and coinversion distributions in Gog triangles.

Abstract

We consider the problem of finding a bijection between the sets of alternating sign matrices and of totally symmetric self complementary plane partitions, which can be reformulated using Gog and Magog triangles. In a previous work we introduced GOGAm triangles, which are images of Magog triangles by the Sch\"utzenberger involution. In this paper we introduce left Gog and GOGAm trapezoids. We conjecture that they are equienumerated, and we give an explicit bijection between such trapezoids with one or two diagonals. We also study the distribution of inversions and coinversions in Gog triangles.