Roger Behrend ; Ilse Fischer ; Matjaz Konvalinka

Diagonally and antidiagonally symmetric alternating sign matrices of odd order
dmtcs:6346 
Discrete Mathematics & Theoretical Computer Science,
April 22, 2020,
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)

https://doi.org/10.46298/dmtcs.6346
Diagonally and antidiagonally symmetric alternating sign matrices of odd order
Authors: Roger Behrend ^{1}; Ilse Fischer ^{2}; Matjaz Konvalinka ^{3}
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Roger Behrend;Ilse Fischer;Matjaz Konvalinka
1 School of Mathematics [Cardiff]
2 Fakultät für Mathematik [Wien]
3 Faculty of Mathematics and Physics [Ljubljana]
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS ASMs) 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 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 oddorder DASASMs is the last to have been proved.