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Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Behrend, Roger E., Fischer, Ilse and Konvalinka, Matjaz 2016. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. Discrete Mathematics and Theoretical Computer Science BC , pp. 131-142.

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Abstract

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) 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)x(2n+1) DASASMs is \prod_{i=0}^n (3i)!/(n+i)!, and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n+1)x(2n+1) 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.

Item Type: Article
Date Type: Published Online
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: Exact enumeration, alternating sign matrices, six-vertex model
Additional Information: Listed as OA on the DOAJ (accessed 18.11.16).
Publisher: Discrete Mathematics and Theoretical Computer Science
ISSN: 1365-8050
Date of First Compliant Deposit: 18 November 2016
Last Modified: 02 May 2019 16:23
URI: http://orca.cf.ac.uk/id/eprint/94629

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