Summary: In the present paper, the Carlitz number is introduced, which is defined by the following recursive formula \[ F(n,k,r)=F(n-1,k-1,r)+F(n-1,k,r- 1)+(q^ k-1)F(n-1,k,r) \] with \(F(0,0,1)=1\), \(F(n,k,0)=0\), if \(n\geq k\geq 0\); \(F(n,k,r)=0\), unless \(0\leq k\leq n\) and \(r\geq 0\). The main results are like this: Theorem 2: Let F(n,k,r) be defined as above. Then, F(n,k,r) counts the number of k-dimensional subspaces S of \(V_ n(q)\) such that S is contained in some \((n-r+1)\)-dimensional coordinate plane but the number \((n-r+1)\) cannot be replaced by (n-r). Theorem 3: Let F(n,k,r) be the same as in theorem 2. Then \[ \left[ \begin{matrix} n\\ k\end{matrix} \right]_ q=\sum_{r\geq 1}F(n,k,r) \]
\[ \left[ \begin{matrix} n\\ k\end{matrix} \right]_ q=\sum^{n}_{j=k+r-1}\left( \begin{matrix} n\\ j\end{matrix} \right)F(j,k,r)/\left( \begin{matrix} \quad n-j+r-1\\ r-1\end{matrix} \right)+\sum^{r-1}_{i=1}F(n,k,i). \]