A bijective proof of the ASM theorem, Part I: the operator formula

TL;DR

This paper presents the first bijective proof of the operator formula for monotone triangles, establishing a key combinatorial link in the study of alternating sign matrices and related objects.

Contribution

It introduces a novel bijective approach using signed sets and sijections to prove the operator formula for monotone triangles.

Findings

First bijective proof of the operator formula

Signed sets and sijections are effective tools

Provides a combinatorial foundation for equivalences

Abstract

Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but no bijective proof for any of these equivalences has been found so far. In this paper we provide the first bijective proof of the operator formula for monotone triangles, which has been the main tool for several non-combinatorial proofs of such equivalences. In this proof, signed sets and sijections (signed bijections) play a fundamental role.