Abstract:
Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, and their numbers are given by a simple product formula, but no bijective proof for any of these results has been found so far. 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. 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. (Joint work with Matjaz Konvalinka.)