Séminaire Lotharingien de Combinatoire, 89B.69 (2023), 12 pp.
Ilse Fischer, Hans Höngesberg and Florian Schreier-Aigner
Alternating Sign Matrices and Descending Plane Partitions: a Linear Number of Equivalent Statistics
Abstract.
There is the same number of alternating sign matrices as there is of cyclically symmetric lozenge tilings of a hexagon with a central triangular hole of size 2, but finding an explicit bijection has been an open problem for about 40 years now. This is even more surprising in the view of the fact that, when restricting to vertically symmetric alternating sign matrices,
their number equals the number of such lozenge tilings that also exhibit an additional reflective symmetry. To approach such bijections, we present generalizations of these results that involve a linear number of equidistributed statistics. Prior to this work, only a constant number of such statistics were known.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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