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