alpha-words and the radix order
Let alpha = (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) is an element of N(n) with 0  = 2. Many interesting combinatorial properties of alpha-words have been studied by Chuan. In this paper, we obtain some new methods of generating the distinct alpha-words of the same order in lexicographic order. Among other results, we consider another function r bar right arrow w[r] from the set of labels of alpha-words to the set of alpha-words. The string r is called a new label of the alpha-word w[r]. Using any new label of an nth-order alpha-word w, we can compute the number of the nth-order alpha-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 is an element of D and r <(r) S. (C) 2010 Elsevier B.V. All rights reserved.
alpha-word radix order lexicographic order