Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)

Extended abstracts listed by author > Fischer Ilse

Diagonally and antidiagonally symmetric alternating sign matrices of odd order
Roger Behrend  1  , Ilse Fischer  2  , Matjaz Konvalinka  3  
1 : Cardiff School of Mathematics [Cardiff]  -  Website
Cardiff School of Mathematics Cardiff University Senghennydd Road, Cardiff, Wales, UK, CF24 4AG -  Royaume-Uni
2 : Fakultät für Mathematik [Wien]  -  Website
Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria -  Autriche
3 : Faculty of Mathematics and Physics [Ljubljana]  (FMF)  -  Website
Jadranska ulica 19, SI-1000 Ljubljana, Slovenia -  Slovénie

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

isn (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. 

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