TY - JOUR

T1 - On the weighted enumeration of alternating sign matrices and descending plane partitions

AU - Behrend, Roger E.

AU - Di Francesco, Philippe

AU - Zinn-Justin, Paul

N1 - Funding Information:
✩ P.D.F. and P.Z.-J. acknowledge partial support from ANR program “GRANMA” BLAN08-1-13695. E-mail addresses: behrendr@cardiff.ac.uk (R.E. Behrend), philippe.di-francesco@cea.fr (P. Di Francesco), pzinn@lpthe.jussieu.fr (P. Zinn-Justin).

PY - 2012/2

Y1 - 2012/2

N2 - 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 -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 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.

AB - 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 -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 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.

KW - Alternating sign matrices

KW - Descending plane partitions

KW - Nonintersecting lattice paths

KW - Six-vertex model with domain-wall boundary conditions

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U2 - 10.1016/j.jcta.2011.09.004

DO - 10.1016/j.jcta.2011.09.004

M3 - Article

AN - SCOPUS:80053609236

SN - 0097-3165

VL - 119

SP - 331

EP - 363

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 2

ER -