  In a recent work, the combinatorial interpretation of the polynomial
alpha(n;k1,k2,...,kn) counting the number of Monotone Triangles with bottom row
k1 < k2 < ... < kn was extended to weakly decreasing sequences k1 >= k2 >= ...
>= kn. In this case the evaluation of the polynomial is equal to a signed
enumeration of objects called Decreasing Monotone Triangles. In this paper we
define Generalized Monotone Triangles - a joint generalization of both ordinary
Monotone Triangles and Decreasing Monotone Triangles. As main result of the
paper we prove that the evaluation of alpha(n;k1,k2,...,kn) at arbitrary
(k1,k2,...,kn) in Z^n is a signed enumeration of Generalized Monotone Triangles
with bottom row (k1,k2,...,kn). Computational experiments indicate that certain
evaluations of the polynomial at integral sequences yield well-known round
numbers related to Alternating Sign Matrices. The main result provides a
combinatorial interpretation of the conjectured identities and could turn out
useful in giving a bijective proof.
