α-words and the radix order

• Authors: Wai-Fong Chuan;Chun-Yu Chen;Hui-Ling Ho;Fang-Yi Liao
• Venue: Theoretical Computer Science
• Year: 2011

Let @a=(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"1r"2...r"n@?N^n, with 0@?r"i@?a"i for 1@?i@?n, there corresponds an nth-order @a-word u"n[r] with label r derived from the pair (a,b). These @a-words are defined recursively as follows: u"-"1=b,u"0=a,u"1[r"1]=a^a^"^1^-^r^"^1ba^r^"^1,u"i[r"1r"2...r"i]=u"i"-"1[r"1r"2...r"i"-"1]^a^"^i^-^r^"^iu"i"-"2[r"1r"2...r"i"-"2]u"i"-"1[r"1r"2...r"i"-"1]^r^"^i,i=2. Many interesting combinatorial properties of @a-words have been studied by Chuan. In this paper, we obtain some new methods of generating the distinct @a-words of the same order in lexicographic order. Among other results, we consider another function r@?w[r] from the set of labels of @a-words to the set of @a-words. The string r is called a new label of the @a-word w[r]. Using any new label of an nth-order @a-word w, we can compute the number of the nth-order @a-words that are less than w in the lexicographic order. With the radix orders