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

Roger E. Behrend, Philippe Di Francesco, Paul Zinn-Justin

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)331-363
Number of pages33
JournalJournal of Combinatorial Theory. Series A
Issue number2
StatePublished - Feb 2012
Externally publishedYes


  • Alternating sign matrices
  • Descending plane partitions
  • Nonintersecting lattice paths
  • Six-vertex model with domain-wall boundary conditions

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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