论文标题
标志变化和下降
Sign variation and descents
论文作者
论文摘要
对于任何$ n> 0 $和$ 0 \ leq m <n $，令$ p_ {n，m} $为$ \ { - ，0，+\} $的投射等价类的姿势 - 长度$ n $的向量，符号变化，由$ m $限制为$ m $，由zeros位置的反向包含命令。令$δ_{n，m} $为$ p_ {n，m} $的订单复合体。第三作者的先前结果表明，$δ_{n，m} $是$ \ mathbb {q} $ cohen-macaulay，每当$ m $均匀或$ m = n-1 $时。因此，因此，$Δ_{n，m} $的$ h $ - 向量由非负条目组成。我们的主要结果指出，$δ_{n，m} $是可以分区的，当$ m $偶数或$ m = n-1 $时，我们对$ h $ - vector进行解释。当$ m = n-1 $时，$ h $ - 向量的条目被证明是Borowiec和Młotkowski在[{\ em Electron中研究的新型欧拉数$ d $。 J. Combin。}，23（1）：纸1.38，13，2016]。然后，我们将主要结果与Klee的广义Dehn-Sommerville关系相结合，以提供有关这些欧拉数$ D $ D $的一些事实的几何证明。
For any $n > 0$ and $0 \leq m < n$, 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 $\mathbb{Q}$ 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 in [{\em Electron. J. Combin.}, 23(1):Paper 1.38, 13, 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 $D$.
