Mathematics – Combinatorics
Scientific paper
2011-03-06
J. Combin. Theory Ser. A 119 (2012) 331-363
Mathematics
Combinatorics
v2: published version
Scientific paper
10.1016/j.jcta.2011.09.004
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1's and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of nxn matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.
Behrend Roger E.
Francesco Philippe Di
Zinn-Justin Paul
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