Enumeration and random generation of accessible automata

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 a set of kn+1 elements into n non-empty subsets. This combinatorial construction shows that the asymptotic order of the cardinality of An is related to the Stirling number . Our bijective approach also yields an efficient random sampler, for the uniform distribution, of automata with n states, its complexity is O(n3/2), using the framework of Boltzmann samplers.