Diagonally Symmetric Alternating Sign Matrices

Roger E. Behrend, Ilse Fischer, Christoph Koutschan

Graphical representation of a DSASM

Abstract

The enumeration of diagonally symmetric alternating sign matrices (DSASMs) is studied, and a Pfaffian formula is obtained for the number of DSASMs of any fixed size, where the entries for the Pfaffian are positive integers given by simple binomial coefficient expressions. This result provides the first known case of an exact enumeration formula for an alternating sign matrix symmetry class in which a simple product formula does not seem to exist. Pfaffian formulae are also obtained for DSASM generating functions associated with several natural statistics, including the number of nonzero strictly upper triangular entries in a DSASM, the number of nonzero diagonal entries in a DSASM, and the column of the unique 1 in the first row of a DSASM. The proofs of these results involve introducing a version of the six-vertex model whose configurations are in bijection with DSASMs of fixed size, and obtaining a Pfaffian expression for its partition function. Various further results and conjectures are also obtained, including some related to the exact enumeration of off-diagonally symmetric alternating sign matrices, and some related to the asymptotic enumeration of DSASMs and other classes of alternating sign matrices.

Paper

The material on this webpage accompanies our article Diagonally Symmetric Alternating Sign Matrices, which is also available on arXiv:2309.08446.

Supplementary electronic material

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