Gog, Magog and Sch\"utzenberger II: Left trapezoids

TL;DR

This paper explores the relationship between Gog and Magog triangles, introducing trapezoids to establish a bijection and conjecturing their equienumeration, advancing understanding of combinatorial objects related to ASMs and TSSCPP.

Contribution

It introduces left Gog and GOGAm trapezoids and provides an explicit bijection for cases with one or two diagonals, proposing a conjecture on their equienumeration.

Findings

Conjecture that Gog and Magog trapezoids are equienumerated.

Explicit bijection established for trapezoids with one or two diagonals.

Enhanced understanding of the combinatorial relationship between Gelfand-Tsetlin triangles, ASMs, and TSSCPP.

Abstract

We are interested in finding an explicit bijection between two families of combinatorial objects: Gog and Magog triangles. These two families are particular classes of Gelfand-Tsetlin triangles and are respectively in bijection with alternating sign matrices (ASM) and totally symmetric self complementary plane partitions (TSSCPP). For this purpose, we introduce left Gog and GOGAm trapezoids. We conjecture that these two families of trapezoids are equienumerated and we give an explicit bijection between the trapezoids with one or two diagonals.