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There is the same number of $n \times n$ alternating sign matrices (ASMs) asthere 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 jointdistribution have been identified. We introduce extensions of ASMs and of DPPsalong with $n+3$ statistics on each extension, and show that the two familiesof statistics have the same joint distribution. The ASM-DPP equinumerosity isobtained as an easy consequence by considering the $(-1)$-enumerations of theseextended objects with respect to one pair of the $n+3$ pairs of statistics. Onemay speculate that the fact that these extensions might be necessary to havethis significance increase in the number of statistics, as well as theinvolvement of signs when specializing to ASMs and DPPs may hint at theobstacles in finding an explicit bijection between ASMs and DPPs. One importanttool for our proof is a multivariate generalization of the operator formula forthe number of monotone triangles with prescribed bottom row that generalizesSchur functions.

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