Sign variation and descents (Q5919070)

Summary: For any \(n > 0\) and \(0 \leqslant m < n\), 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 \(\Delta_{n,m}\) be the order complex of \(P_{n,m}\). A previous result from the third author [``Boundary measurement and sign variation in real projective space'', Preprint, \url{arXiv:1909.04640}] shows that \(\Delta_{n,m}\) is Cohen-Macaulay over \(\mathbb{Q}\) 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 \textit{A. Borowiec} and \textit{W. Młotkowski} [Electron. J. Comb. 23, No. 1, Research Paper P1.38, 13 p. (2016; Zbl 1382.05004)]. 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 \(D\).