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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