TY - JOUR

T1 - A doubly-refined enumeration of alternating sign matrices and descending plane partitions

AU - Behrend, Roger E.

AU - Di Francesco, Philippe

AU - Zinn-Justin, Paul

N1 - Funding Information:
✩ PDF and PZJ acknowledge partial support from ANR program “GRANMA” BLAN08-1-13695. PZJ is supported in part by ERC grant 278124 “LIC”. 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 - 2013/2

Y1 - 2013/2

N2 - 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.

AB - 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.

KW - Alternating sign matrices

KW - Descending plane partitions

KW - Desnanot-Jacobi identity

KW - Nonintersecting lattice paths

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

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U2 - 10.1016/j.jcta.2012.09.004

DO - 10.1016/j.jcta.2012.09.004

M3 - Article

AN - SCOPUS:84867113799

SN - 0097-3165

VL - 120

SP - 409

EP - 432

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 2

ER -