Abstract:

We will start by introducing parallelogram polyominoes, labeled polyominoes, and symmetric functions refining the q-enumeration of these objects by the area statistic.
We will then introduce some symmetric function operators coming from the theory of modified Macdonald polynomials, including Delta, Theta, and the super Nabla operator. One of the surprising facts is that these operators, when specialized, give the q-enumerations that we will describe during the beginning of the talk.
This leads to many open questions regarding the (q,t)-enumeration of these polyominoes.
It is worth mentioning that these expressions and operators have ties to other areas of mathematics, such as knot/link invariants and flag Hilbert schemes, though our focus will be purely combinatorial.
This talk will be based on joint work with A. Iraci, and joint work with F. Bergeron, J. Haglund, and A. Iraci.