Séminaire Lotharingien de Combinatoire, 85B.45 (2021), 12 pp.

Nantel Bergeron, Aram Dermenjian and John Machacek

Sign Variation and Descents

Abstract. 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 [Electron. J. Combin., 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.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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