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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 *P*_{n,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 Δ_{n,m} be the order complex of *P*_{n,m}.
A previous result from the third author shows that Δ_{n,m} is Cohen-Macaulay over **Q** whenever *m* is even or *m* = *n*-1.
Hence, it follows that the *h*-vector of Δ_{n,m} consists of nonnegative entries.
Our main result states that Δ_{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 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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