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. 131142.

Abstract
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the sixvertex 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 YangBaxter 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 oddorder 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, sixvertex model 
Additional Information: 
Listed as OA on the DOAJ (accessed 18.11.16). 
Publisher: 
Discrete Mathematics and Theoretical Computer Science 
ISSN: 
13658050 
Date of First Compliant Deposit: 
18 November 2016 
Last Modified: 
02 May 2019 16:23 
URI: 
http://orca.cardiff.ac.uk/id/eprint/94629 
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