A bijective proof of the ASM theorem, Part II: ASM enumeration and ASM-DPP relation

TL;DR

This paper provides a bijective proof for the enumeration of alternating sign matrices and their equinumerosity with descending plane partitions, refining the bijections with positional information of entries.

Contribution

It introduces a new combinatorial bijective proof for ASM enumeration and ASM-DPP relation, including positional refinements, based on signed sets and computational implementations.

Findings

Established a bijective proof for ASM enumeration formula

Proved ASM are equinumerous with descending plane partitions

Provided computer code for the bijective constructions

Abstract

This paper is the second in a series of planned papers which provide first bijective proofs of alternating sign matrix results. Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for alternating sign matrices and of the fact that alternating sign matrices are equinumerous with descending plane partitions. We are also able to refine these bijections by including the position of the unique $1$ in the top row of the matrix. Our constructions rely on signed sets and related notions. The starting point for these constructions were known ``computational'' proofs, but the combinatorial point of view led to several drastic modifications. We also provide computer code where all of our constructions have been implemented.