Sign Variation and Descents

For any n>0 and 0≤m<n, let Pn,m be the poset of projective equivalence classes of {−,0,+}-vectors of length n with sign variation bounded by m, ordered by reverse inclusion of the positions of zeros. Let Δn,m be the order complex of Pn,m. A previous result from the third author shows that Δn,m is Cohen-Macaulay over Q whenever m is even or m=n−1. Hence, it follows that the h-vector of Δn,m consists of nonnegative entries. Our main result states that Δn,m is partitionable and we give an interpretation of the h-vector when m is even or m=n−1. When m=n−1 the entries of the h-vector turn out to be the new Eulerian numbers of type D studied by Borowiec and Młotkowski in [ Electron. J. Combin., 23(1):#P1.38, 2016]. We then combine our main result with Klee's generalized Dehn-Sommerville relations to give a geometric proof of some facts about these Eulerian numbers of type D.