UW Combinatorics and Geometry Seminar: Galen Dorpalen-Barry

Motivated by a conjecture of Maglione–Voll from group theory, we introduce and study the Poincaré-extended ab-index. This polynomial generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings (a relaxation of EL-labelings), we prove that the coefficients are nonnegative and give a combinatorial description of the coefficients. This proves Maglione–Voll’s conjecture as well as a conjecture of the Kühne–Maglione. We also define the pullback ab-index generalizing the cd-index of face posets for oriented matroids. Our results recover, generalize and unify results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg. This is joint work with Joshua Maglione and Christian Stump.