Roger E. Behrend
Published 2012-03-14, updated 2013-10-30Version 3
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 rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. 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.
Comments: 62 pages; v3 slightly updated relative to published version
Journal: Adv. Math. 245 (2013) 439-499
The relation between alternating sign matrices and descending plane partitions: $n+3$ pairs of equivalent statistics
An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices
Alternating sign matrices and totally symmetric plane partitions
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