Characterizing Alternating Sign Triangles

TL;DR

This paper characterizes permutation triangles within alternating sign triangles, providing a simple criterion and exploring properties related to their structure, advancing understanding in combinatorial matrix theory.

Contribution

It proves a conjecture by Glick that characterizes permutation triangles among alternating sign triangles, and investigates their properties.

Findings

Permutation triangles are characterized by a simple criterion.

The paper confirms Glick's conjecture about permutation triangles.

Properties of alternating sign triangles are established.

Abstract

Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are either 0 or 1, by analogy with permutation matrices. In this paper, we prove a simple characterization of permutation triangles, originally conjectured by Glick. We will also prove some properties of alternating sign triangles.