Combinatorial reciprocity for Monotone Triangles

The number of Monotone Triangles with bottom row k 1 < k 2 < ⋯ < k n is given by a polynomial α ( n ; k 1 , … , k n ) in n variables. The evaluation of this polynomial at weakly decreasing sequences k 1 ⩾ k 2 ⩾ ⋯ ⩾ k n 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 ( 2 n ; 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.