α-words and the radix order

Abstract

a b s t r a c tLet α = (a 1 , a 2 , . . .) be a sequence (finite or infinite) of integers with a 1 ≥ 0 and a n ≥ 1, for all n ≥ 2. Let {a, b} be an alphabet. For n ≥ 1, and r = r 1 r 2 · · · r n ∈ N n , with 0 ≤ r i ≤ a i for 1 ≤ i ≤ n, there corresponds an nth-order α-word u n [r] with label r derived from the pair (a, b). These α-words are defined recursively as follows:Many interesting combinatorial properties of α-words have been studied by Chuan. In this paper, we obtain some new methods of generating the distinct α-words of the same order in lexicographic order. Among other results, we consider another function r  → w[r] from the set of labels of α-words to the set of α-words. The string r is called a new label of the α-word w[r].Using any new label of an nth-order α-word w, we can compute the number of the nthorder α-words that are less than w in the lexicographic order. With the radix orders < r on N n (regarding N as an alphabet) and {a, b} + with a < r b, we prove that there exists a subset D of the set of all labels such that w[r] < r w[s] whenever r, s ∈ D and r < r s.