1203.3187 (Roger E. Behrend)

The quadruply-refined enumeration of alternating sign matrices    [PDF]

Roger E. Behrend

A formula is obtained for a six-parameter generating function for alternating sign matrices (ASMs) of fixed size. The parameters are associated with the four natural boundary statistics for an ASM, i.e., the positions of the 1's in the first and last row and column, and two further bulk statistics for an ASM, specifically the numbers of generalized inversions and -1's. The derivation involves applying the Desnanot-Jacobi determinant identity to the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions. The formula enables the six-parameter generating function to be expressed entirely in terms of previously-known functions or determinants.

View original: http://arxiv.org/abs/1203.3187