We present a bijection between the set $\mathcal{A}_n$ of deterministic and accessible automata with $n$ states on a $k$-letters alphabet and some diagrams, which can themselves be represented as partitions of the set $[\![ 1..(kn+1) ]\!]$ into $n$ non-empty parts. This combinatorial construction shows that the asymptotic order of the cardinality of $\mathcal{A}_n$ is related to the Stirling number $\{^{kn}_n\}$. Our bijective approach also yields an efficient random sampler of automata with $n$ states, of complexity $O(n^{3/2})$, using the framework of Boltzmann samplers.