Intersecting antichains and shadows in linear lattices
Journal of Combinatorial Theory Series A
| Hi-index | 0.00 |
Let $V$ be the $n$-dimensional vector space over the finite field with $q$ elements and $L(V)$ the lattice of subspaces of $V$ ordered by inclusion. Let $V_1,V_2,\ldots,V_r$ be $r$ selected subspaces of $V$ such that $\{0\}=V_0 \subseteq V_1 \subseteq \cdots \subseteq V_r= V$ and $\dim(V_i)=\sum _{j=1}^{i}k_j$. Let ${\cal I}_i$ be a subinterval of $[0,k_i]$, $i=1,\ldots,r$, let $I=\{{\cal I}_1,\ldots,{\cal I}_r\}$, and let $C[n,r,I]=\{K \in L(V): \dim (K \cap V_i)-\dim (K \cap V_{i-1})\in {\cal I}_i, i=1,2,\ldots,r\}$. Then $C[n,r,I]$ is a graded poset. In this paper, using group actions and flow morphisms we show that $C[n,r,I]$ is log concave and has the NM property, which yields that $C[n,r,I]$ has the strong Sperner property and the LYM property.