A formula for a doubly refined enumeration of alternating sign matrices

Zeilberger (1996) [12] proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey (1983) [9], for the number of alternating sign matrices with given top row. Stroganov (2006) [10] proved an explicit formula for the number of alternating sign matrices with given top and bottom rows. Fischer and Romik (2009) [7] considered a different kind of "doubly-refined enumeration" where one counts alternating sign matrices with given top two rows, and obtained partial results on this enumeration. In this paper we continue the study of the doubly-refined enumeration with respect to the top two rows, and use Stroganov's formula to prove an explicit formula for these doubly-refined enumeration numbers. (C) 2009 Elsevier Inc. All rights reserved.