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

Behrend, Roger E. ORCID: https://orcid.org/0000-0002-6143-7439, 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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Abstract

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
URI:	https://orca.cardiff.ac.uk/id/eprint/13208

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