We enumerate the alternating sign matrices that contain exactly one -1 according to their number of inversions (possibly taking into account the position of the unique non-zero entry in the first row). In conformity with the Mills, Robbins and Rumsey conjectures, this is the same as the enumeration, according to the number of parts, of descending plane partitions with exactly one special part. This is shown by finding a determinantal expression for the generating function of descending plane partitions, transforming it algebraically and extracting recurrences for those with one special part. Finally, we show that the generating function of alternating sign matrices that contain exactly one -1 follows the same recurrences. © 2002 Elsevier Science B.V. All rights reserved.
Lalonde, P. (2002). q-Enumeration of alternating sign matrices with exactly one -1. Discrete Mathematics, 256(3), 759–773. https://doi.org/10.1016/S0012-365X(02)00346-1