Aspects of enumeration and generation with a string automata representation

In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata (ICDFAs) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFAs but, to some extent, the set of regular languages. An exact generation algorithm can be used to partition the set of ICDFAs in order to parallelize the counting of minimal automata (and thus of regular languages). We present also a uniform random generator for ICDFAs that uses a table of pre-calculated values. Based on the same table it is also possible to obtain an optimal coding for ICDFAs.