Subspaces, subsets, and Motzkin paths

Abstract

Let $M\left(n\right)$ denote the set of all Motzkin paths from $(0,0)$ to $(n,0)$. For each $P\in M\left(n\right)$ we define a statistic $w(P,q)$, the weight of $P$. Let $\left|P\right|$ denote the number of down steps in $P\in M\left(n\right)$. Let $B_q\left(n\right)$ denote the projective geometry (= poset of subspaces of an $n$-dimensional vector space over $F_q$). We define a map from $B_q\left(n\right)$ to $M\left(n\right)$ and show that, for $P\in M\left(n\right)$, the inverse image of $P$ consists of a disjoint union of ${(q-1)}^{\left|P\right|}w(P,q)$ symmetric Boolean subsets in $B_q\left(n\right)$, all with minimum rank $\left|P\right|$ and maximum rank $n-\left|P\right|$. This yields an explicit symmetric Boolean decomposition of the projective geometry and gives a poset theoretic interpretation to the identity ${\left(\frac{n}{k}\right)}_q=\sum _{P\in M\left(n\right)}{(q-1)}^{\left|P\right|}w(P,q)\left(\frac{n-2\left|P\right|}{k-\left|P\right|}\right).$