Conjecture on monotone triangle evaluations and vertically symmetric alternating sign matrices

Let α(n;k1,,kn)\alpha(n;k_1,\ldots,k_n) denote the number of monotone triangles with bottom row (k1,,kn)(k_1,\ldots,k_n), extended to arbitrary integer bottom rows by the associated evaluation. For n=2m+1n=2m+1 with m1m\geq 1, consider the evaluations with bottom rows (n,n1,,1)(n,n-1,\ldots,1) and (2,4,,2m)(2,4,\ldots,2m). The monotone triangle evaluation conjecture.

α(n;n,n1,,1)=(1)mα(m;2,4,,2m).\alpha(n;n,n-1,\ldots,1)=(-1)^m\alpha(m;2,4,\ldots,2m).

The absolute values of the left-hand evaluations are the numbers of vertically symmetric alternating sign matrices of odd order, while the right-hand side counts monotone triangles with prescribed bottom row (2,4,,2m)(2,4,\ldots,2m). The displayed identity is supported by the initial computed values, but its status is not resolved in the supplied text.

Sources & referencesView supporting material

Primary source

Ilse Fischer and Lukas Riegler, “Combinatorial Reciprocity for Monotone Triangles”, arXiv:1111.2695 (2011).

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