The relation between alternating sign matrices and descending plane partitions: $n+3$ pairs of equivalent statistics

TL;DR

This paper introduces extended versions of alternating sign matrices and descending plane partitions with $n+3$ statistics, showing they share the same joint distribution, which sheds light on the longstanding open problem of explicit bijections.

Contribution

It extends ASMs and DPPs with additional statistics and proves their joint distributions match, advancing understanding of their combinatorial relationship.

Findings

Same joint distribution of extended ASMs and DPPs with $n+3$ statistics

Equinumerosity derived from $(-1)$-enumerations of extended objects

Multivariate operator formula generalizes Schur functions

Abstract

There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is an open problem for about $40$ years now. So far, quadruples of statistics on ASMs and on DPPs that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with $n+3$ statistics on each extension, and show that the two families of statistics have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the $(-1)$-enumerations of these extended objects with respect to one pair of the $n+3$ pairs of statistics. One may speculate that the fact that these extensions might be necessary to have this significance increase in the number of statistics, as well as the involvement of signs when specializing to ASMs and DPPs may hint at the obstacles in finding an explicit bijection between ASMs and DPPs. One important tool for our proof is a multivariate generalization of the operator formula for the number of monotone triangles with prescribed bottom row that generalizes Schur functions.