Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order

Abstract

For each $α\in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $α$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vertically and horizontally symmetric ASMs. We also prove several refinements. For instance, in the case of DASASMs with a maximal number of $-1$'s along the diagonal and the antidiagonal, these considerations lead naturally to the definition of alternating sign triangles. These are new objects that are equinumerous with ASMs, and we are able to prove a two parameter refinement of this fact, involving the number of $-1$'s and the inversion number on the ASM side. To prove our results, we extend techniques to deal with triangular six-vertex configurations that have recently successfully been applied to settle Robbins' conjecture on the number of all DASASMs of odd order. Importantly, we use a general solution of the reflection equation to prove the symmetry of the partition function in the spectral parameters. In all of our cases, we derive determinant or Pfaffian formulas for the partition functions, which we then specialize in order to obtain the product formulas for the various classes of extreme odd DASASMs under consideration.