Let be a sequence (finite or infinite) of integers with and , for all . Let be an alphabet. For , and , with for , there corresponds an th-order -word with label derived from the pair . These -words are defined recursively as follows: Many interesting combinatorial properties of -words have been studied by Chuan. In this paper, we obtain some new methods of generating the distinct -words of the same order in lexicographic order. Among other results, we consider another function from the set of labels of -words to the set of -words. The string is called a new label of the -word .
Using any new label of an th-order -word , we can compute the number of the th-order -words that are less than in the lexicographic order. With the radix orders on (regarding as an alphabet) and with , we prove that there exists a subset of the set of all labels such that whenever and .