A doubly-refined enumeration of alternating sign matrices and descending plane partitions

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

Research output: Contribution to journalArticlepeer-review

Abstract

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on n × n alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of -1's in A and the number of 0's to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of n's in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0's to the right of the 1 in the last row of an ASM, and the number of (n - 1)'s plus the number of rows of length n - 1 in a DPP. This generalization is proved using the known equality of the three-statistic generating functions, together with relations which express each four-statistic generating function in terms of its three-statistic counterpart. These relations are obtained by applying the Desnanot-Jacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the six-vertex model with domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.

Original languageEnglish (US)
Pages (from-to)409-432
Number of pages24
JournalJournal of Combinatorial Theory. Series A
Volume120
Issue number2
DOIs
StatePublished - Feb 2013
Externally publishedYes

Keywords

  • Alternating sign matrices
  • Descending plane partitions
  • Desnanot-Jacobi identity
  • 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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