State complexity of permutation on finite languages over a binary alphabet

Abstract The set of all strings Parikh equivalent to a string in a language L is called the permutation of L . The permutation of a finite n -state DFA (deterministic finite automaton) language over a binary alphabet can be recognized by a DFA with n 2 − n + 2 2 states. We show that if the language consists of equal length binary strings the bound can be improved to f ( n ) = n 2 + n + 1 3 and for every n congruent to 1 modulo 3 there exists an n -state DFA A recognizing a set of equal length strings such that the minimal DFA for the permutation of L ( A ) needs f ( n ) states.