Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last row and column of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known results for the exact enumeration of ASMs with prescribed values of some of these statistics are reviewed in detail. A quadratic relation which fully determines the generating function associated with all six statistics is then obtained. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions