A finite field approach to the Carlitz–Riordan q-Catalan numbers

Abstract

Let q be a prime power and ${\mathbb {F}}_q$ be the finite field with q elements. Suppose that $n\ge 1$ and${{\mathscr {F}}}=\{E_1,\ldots ,E_{2n-1}\}$ is a collection of subspaces of ${\mathbb {F}}_q^{2n}$ with $E_i\subseteq E_{i+1}$ and $\dim E_i=i$. We prove that

$$\begin{aligned} |\{V\subseteq {\mathbb {F}}_q^{2n}:\, \dim V=n,\ \dim (V\cap E_i)\ge i/2 {\text { for }}1\le i<2n\}|=C_n(q), \end{aligned}$$

where the Carlitz–Riordan q-Catalan number $C_n(q)$ is given by

$$\begin{aligned} C_0(q)=1,\quad C_n(q)=\sum _{k=0}^{n-1}C_k(q)C_{n-1-k}(q)q^{(k+1)(n-1-k)}. \end{aligned}$$