# A conjecture on the Hilbert series of binomial ideals associated with simple polyominoes

# Dr. Ayesha Qureshi

Sabanci University

## 13 May 2022 17:00

## Abstract

We will discuss certain types of binomial ideals arising from combinatorial struc-
tures. In particular, focus will be on the binomial ideals arising from

(1) certain sets of 2-minors of a matrix of indeterminates (also known as poly-
omino ideals);

(2) incomparable elements in finite distributive lattices (also known as join-meet
ideals).
We will present a conjecture on the reduced Hilbert series of the coordinate ring
of a simple polyomino ideal in terms of particular arrangements of non-attacking
rooks that can be placed on the polyomino. By using a computational approach,
the conjecture holds for all simple polyominoes up to rank 11. By using an alge-
braic approach, the conjecture holds true for the class of parallelogram polyomino
ideals, by looking at those as join-meet ideals of simple planar distributive lattices.

We will also give a combinatorial interpretation(in terms of Motzkin paths) of the Gorensteinnes of parallelogram polyomino ideals.
This talk is based on a recent joint work with Francesco Romeo and Giancarlo
Rinaldo.

References:

1) A.A.Qureshi, F. Romeo, G. Rinaldo, ”Hilbert series of Parallelogram Polyomi-
noes”, arXiv:2111.01907