On the enumerative sequences of regular languages on k symbols

Abstract

The main result is a characterization of enumerative sequences of regular languages on k symbols. We prove that a sequence is the generating series s(z) of a regular language on k symbols if and only if it is the generating series of a language over a k-letter alphabet and if both series s(z) and (kz)∗ - s(z) are regular. The proof uses transformations on linear representations called inductions.