Combinatorics of diagrams of permutations

11/15/2016 - 12:15pm
11/15/2016 - 1:10pm
Alejandro Morales (UCLA)

There are numerous combinatorial objects associated to a Grassmannian permutation w_\lambda that indexes cells of the totally nonnegative Grassmannian. We study several of these objects and their q-analogues in the case of permutations w that are not necessarily Grassmannian. We give two main results: first, we show that certain regions of an inversion arrangement, rook placements avoiding a diagram of w, and fillings of a diagram of w are equinumerous for all permutations w. Second, we give a q-analogue of these results by showing that the number of invertible matrices over a finite field avoiding a diagram of w is a polynomial in the size of the field, and for certain permutations a polynomial with nonnegative coefficients. This is joint work with Joel Lewis. The talk will be accessible to undergraduates familiar with discrete math.

Millikan 2099, Pomona College