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  1. Home > Articles & Issues >
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Conference paper

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Roger Behrend (1), Ilse Fischer ORCID (2), Matjaz Konvalinka ORCID (3)
(1) School of Mathematics [Cardiff]
(2) Fakultät für Mathematik [Wien]
(3) Faculty of Mathematics and Physics [Ljubljana]
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July 4, 2016
Published on
April 22, 2020
Last modified on
March 31, 2025
Volume
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
DOI
10.46298/dmtcs.6346
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Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Roger Behrend (1), Ilse Fischer ORCID (2), Matjaz Konvalinka ORCID (3)
(1) School of Mathematics [Cardiff]
(2) Fakultät für Mathematik [Wien]
(3) Faculty of Mathematics and Physics [Ljubljana]
Abstract
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.
Keywords
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
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