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 non-empty 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(n3/2), using the framework of Boltzmann samplers. finite automata, bijection, asymptotic enumeration, random generation, Boltzmann samplers