Sign Variation And Descents

For any 77, > 0 and 0 <= m < n, let P-n,P-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 Delta(n,m) be the order complex of P-n,P-m. A previous result from the third author shows that Delta(n,m) is Cohen-Macaulay over Q whenever m. is even or m = n-1. Hence, it follows that the h-vector of Delta(n,m) consists of nonnegative entries. Our main result states that Delta(n,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 Mlotkowski 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.