Alternating sign matrices and totally symmetric plane partitions

Florian Aigner , Ilse Fischer , Matjaž Konvalinka (1) , Philippe Nadeau (2) , Vasu Tewari (3)

1 University of Ljubljana - Department of Mathematics
2 CNRS - Centre National de la Recherche Scientifique
3 University of Pennsylvania

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.

Mots clés: alternating sign matrices; totally symmetric plane partitions; Schur polynomials; Catalan numbers; alternating sign matrices

Domaines: Combinatoire [math.CO]

Florian Aigner, Ilse Fischer, Matjaž Konvalinka, Philippe Nadeau, Vasu Tewari. Alternating sign matrices and totally symmetric plane partitions. Seminaire Lotharingien de Combinatoire, 2020. ⟨hal-03093904⟩