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Combinatorial Reciprocity for Monotone Triangles

Abstract : The number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ is given by a polynomial $\alpha (n; k_1,\ldots,k_n)$ 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 $\alpha (n;1,2,\ldots,n) = \alpha (2n; n,n,n-1,n-1,\ldots,1,1)$. In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row $(n,n,n-1,n-1,\ldots,1,1)$ 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.
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Ilse Fischer, Lukas Riegler. Combinatorial Reciprocity for Monotone Triangles. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 2012, Nagoya, Japan. pp.313-324, ⟨10.46298/dmtcs.3042⟩. ⟨hal-01283173⟩

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