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On the weighted enumeration of alternating sign matrices and descending plane partitions

Behrend, Roger E. ORCID:, Di Francesco, Philippe and Zinn-Justin, Paul 2012. On the weighted enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 119 (2) , pp. 331-363. 10.1016/j.jcta.2011.09.004

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We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340–359] that, for any n, k, m and p, the number of n×n alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m −1ʼs 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 n×n 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 Lindström–Gessel–Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: Alternating sign matrices; Descending plane partitions; Six-vertex model with domain-wall boundary conditions; Nonintersecting lattice paths
Publisher: Elsevier
ISSN: 0097-3165
Last Modified: 09 May 2023 03:34

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