More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general "d-refined" enumerations of ASMs according to the first d rows. For the doubly-refined case of d = 2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A(n,i,j) that enumerate such matrices. We give a conjectural explicit formula for A(n,i,j) and formulate several other conjectures about the sufficiency of the linear equations to determine the A(n,i,j)'s and about an extension of the linear equations to the general d-refined enumerations. (C) 2009 Elsevier Inc. All rights reserved.