α-words and the radix order

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Abstract

Let α=(a1,a2,) be a sequence (finite or infinite) of integers with a10 and an1, for all n2. Let {a,b} be an alphabet. For n1, and r=r1r2rnNn, with 0riai for 1in, there corresponds an nth-order α-word un[r] with label r derived from the pair (a,b). These α-words are defined recursively as follows: u1=b,u0=a,u1[r1]=aa1r1bar1,ui[r1r2ri]=ui1[r1r2ri1]airiui2[r1r2ri2]ui1[r1r2ri1]ri,i2. 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 rw[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 nth-order α-words that are less than w in the lexicographic order. With the radix orders <r on Nn (regarding N as an alphabet) and {a,b}+ with a<rb, we prove that there exists a subset D of the set of all labels such that w[r]<rw[s] whenever r,sD and r<rs.

Keywords

α-word
Radix order
Lexicographic order

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