A Bijective Proof of the ASM Theorem, Part I: The Operator Formula

Ilse Fischer,Matjaž Konvalinka

doi:10.37236/9082

Ilse Fischer, Matjaž Konvalinka

Open Access

https://doi.org/10.37236/9082

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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.

Highlights

An alternating sign matrix (ASM) is a square matrix with entries in {0, 1, −1} such that in each row and each column the non-zero entries alternate and sum to 1
The operator formula is the main tool for non-combinatorial proofs of several results where alternating sign matrix objects are related to plane partition objects, or for showing that n × n ASMs are enumerated by [1]
The operator formula was used in [Fis07] to show that n × n ASMs are counted by [1] and, more generally, to count ASMs with respect to the position of the unique 1 in the top row. While working on this project, we realized that the final calculation in [Fis07] implies that ASMs are equinumerous with descending plane partitions (DPPs) without having to use Andrews’ result [And79] on the number of DPPs; more generally, we can even prove that the refined count of n × n ASMs with respect to the position of the unique 1 in the top row agrees with the refined count of DPPs with parts no greater than n with respect to the number of parts equal to n

Summary

Introduction

An alternating sign matrix (ASM) is a square matrix with entries in {0, 1, −1} such that in each row and each column the non-zero entries alternate and sum to 1. In very recent work, alternating sign the electronic journal of combinatorics 27(3) (2020), #3.35 triangles (ASTs) were introduced in [ABF20], which establishes a fourth class of objects that are equinumerous with ASMs, and in this case nobody has so far been able to construct a bijection. Another aspect that should be mentioned here is Okada’s work [Oka06](see [Str]), which hints at a connection between ASMs and representation theory that has not yet been well understood.

Signed sets and sijections

Some sijections on signed boxes

Gelfand–Tsetlin patterns

Combinatorics of the monotone triangle recursion

Future work