Appears in collection : Beyond Permutahedra and Associahedra / Au-dela du Permutoèdre et de l'associaèdre

I introduce the extended ab-index of a hyperplane arrangement and discuss some of its properties. I will (1) show its connection to the cd-index and to the ab-index of its face poset, (2) discuss its evaluations to the coarse flag Hilbert-Poincaré series and to the Chow polynomial, (3) present a companion statistic for the descents on permutations that yields the extended ab-index, and (4) end with the open problem to find such a companion statistic for reflection arrangements that would yield a long-sought generalization of left-to-right maxima on permutations to finite reflection groups. Based on joint works with Lorenzo Vecchi, Galen Dorpalen-Barry, Elena Hoster, and Josh Maglione.