Combinatorial reciprocity for Monotone Triangles

Abstract

The number of Monotone Triangles with bottom row k1<k2<⋯<kn is given by a polynomial α(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 α(n;1,2,…,n)=α(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.