Alternating Sign Matrices and Descending Plane Partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1,-1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math. 53 (1979), 193-225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Znuent. Math. 66 (1982), 73-87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented. 1. DEFINITIONS We begin with a definition. DEFINITION 1. An alternating sign matrix is a square matrix which satisfies: (i) all entries are 1,-1, or 0, (ii) every row and column has sum 1, (iii) in every row and column the nonzero entries alternate in sign. All permutation matrices are alternating sign matrices. For 1 x 1 and 2 x 2 matrices these are the only alternating sign matrices. There are exactly seven alternating sign 3 X 3 matrices, six permutation matrices and the matrix 340