Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Roger E Behrend (1) , Ilse Fischer (2) , Matjaz Konvalinka (3)

1 School of Mathematics [Cardiff]
2 Fakultät für Mathematik [Wien]
3 FMF - 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 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 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 odd-order DASASMs is the last to have been proved.

Domaines: Combinatoire [math.CO]

Roger E Behrend, Ilse Fischer, Matjaz Konvalinka. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6346⟩. ⟨hal-02173748⟩