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Séminaire Lotharingien de Combinatoire, 84B.77 (2020), 12 pp.

# Florian Aigner, Ilse Fischer, Matjaž Konvalinka, Philippe Nadeau, and Vasu Tewari

# Alternating Sign Matrices and Totally Symmetric Plane Partitions

**Abstract.**
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the unique 1 in the top row.
We prove that the expansion can be expressed as a sum over totally symmetric plane partitions and we are also able to determine the coefficients.
This establishes a new connection between alternating sign matrices and a class of plane partitions, thereby complementing the fact that
alternating sign matrices are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. As a by-product we obtain an interesting map from totally symmetric plane partitions to Dyck paths. The proof is based on a new, quite general antisymmetrizer-to-determinant formula.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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