Abstract: We present a new family of Schur positive symmetric polynomials, which are defined as sums over totally symmetric plane partitions. In the first part of the talk, we show that these symmetric polynomials generalise a multivariate generating function of alternating sign matrices. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. In the second part, we relate three specialisations of these symmetric polynomials to a weighted enumeration of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.
This is joint work with Ilse Fischer.