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 > Behrend Roger

Friday 8
Andrew Rechnitzer
› 10:00 - 10:30 (30min)
› Djavad Mowafaghian Cinema
Diagonally and antidiagonally symmetric alternating sign matrices of odd order
Roger Behrend  1  , Ilse Fischer  2  , Matjaz Konvalinka  3  
1 : School of Mathematics, Cardiff University
CF24 4AG Cardiff -  Royaume-Uni
2 : Fakultät für Mathematik, Universitat Wien  -  Website
Oskar-Morgenstern-Platz 1, 1090 Wien -  Autriche
3 : Fakulteta za matematico in fiziko, Univerza v Ljubljani in Institut za matematiko
fiiziko inmehaniko, Jadranska 19, Ljubljana -  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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