Enumeration and random generation of accessible automata
Résumé
We present a bijection between the set 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 a set of kn+1 elements into n non-empty subsets. This combinatorial construction shows that the asymptotic order of the cardinality of A_n is related to Stirling numbers of the second kind . Our bijective approach also yields an efficient random sampler of automata with n states for the uniform distribution: its complexity is O(n^{3/2}), using the framework of Boltzmann samplers.
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