We present a bijection between the set An 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 nonempty parts. This combinatorial construction shows that the asymptotic order of the cardinality of An 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.