  The number of Monotone Triangles with bottom row k1 < k2 < ... < kn is given
by a polynomial alpha(n; k1,...,kn) in n variables. The evaluation of this
polynomial at weakly decreasing sequences k1 >= k2 >= ... >= kn turns out to be
interpretable as signed enumeration of new combinatorial objects called
Decreasing Monotone Triangles. There exist surprising connections between the
two classes of objects -- in particular it is shown that alpha(n; 1,2,...,n) =
alpha(2n; n,n,n-1,n-1,...,1,1). In perfect analogy to the correspondence
between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing
Monotone Triangles with bottom row (n,n,n-1,n-1,...,1,1) is in one-to-one
correspondence with a certain set of ASM-like matrices, which also play an
important role in proving the claimed identity algebraically. Finding a
bijective proof remains an open problem.
