Alternating sign matrices and totally symmetric plane partitions
Abstract
We introduce a new family $\mathcal{A}_{n,k}$ of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for $k=1$, this family is equal to a multivariate generating function involving $n+3$ variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric selfcomplementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizertodeterminant formula for which we also provide a bijective proof. In the second part, we relate three specialisation of $\mathcal{A}_{n,k}$ to a weighted enumeration of certain wellknown classes of column strict shifted plane partitions that generalise descending plane partitions.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.13142
 Bibcode:
 2022arXiv220113142A
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 05A05;
 05A15;
 05E05;
 15B35;
 82B20;
 82B23