α-Words and Lexicographic Order

• Main Authors: Chun-Yu Chen, 陳桾瑀
• Other Authors: Wai-Fong Chuan
• Format: Others
• Language: en_US
• Published: 2008
• Online Access: http://ndltd.ncl.edu.tw/handle/32646189443182080683

碩士 === 中原大學 === 應用數學研究所 === 96 === Let α = {a1, a2, ...} be a sequence (finite or infinite) of nonnegative integers with a1  0 and an  1, for all n  2. Let A = {a, b} be an alphabet. For n  1, and r1r2 ...rn 2 Nn, with 0 · ri · ai for 1 · i · n, there associates an nth-order α-word un[r1r2 ...rn] derived from the pair (a, b). These α-words are defined recursively as follows: u1 = b, u0 = a, u1[r1] = aa1r1bar1 and ui[r1r2 ¢ ¢ ¢ ri] = ui1[r1r2 ...ri1]airiui2[r1r2 ...ri2]ui1[r1r2 ...ri1]ri , i  2. We call r1r2 ...rn the label of the word un[r1r2...rn]. a, b are α-word with empty label. The properties of α-words have been studied extensively. Some results on the lexicographic order for the nth-order α-word are known. In this thesis, we prove some new results on the lexicographic order of the nth-order α-words. We find a new method to generate the α-words by defining w1 = b, w0 = a, w1[r1] = aa1r1bar1 , w2[r1r2] = w1[r1]r2aw1[r1]a2r2 and for n  3, wn[r1r2 ¢ ¢ ¢ rn] = wn1[r1r2 ...rn1]anrnwn2[r1r2...rn2]wn1[r1r2 ...rn1]rn if n is odd, wn1[r1r2 ...rn1]rnwn2[r1r2... rn2]wn1[r1r2 ....rn1]anrn i f n is even. We show that each wn[r1r2 ...rn] is an α-word. We construct a tree T associated with the set of α-words: b is the root (the leftmost vertex) of T, a is the only son of b, w1[0], w1[1],...,w1[a1] are the sons of a and wn+1[r1r2 ...rn0], wn+1[r1r2 ¢ ¢ ¢ rn1],... , wn+1[r1r2 ...rnan+1] are the an+1 + 1 sons of wn[r1r2... rn], listing from top to bottom. Label the edge (wn[r1r2...rn], wn+1[r1r2 ...rnrn+1]) by rn+1, n  0. The new label r1r2 ¢ ¢ ¢ rn of the α-word wn[r1r2 ¢ ¢ ¢ rn] represents a path from the root b to the vertex wn[r1r2...rn]. With the lexicographic order on N+ and A+ with a < b, we prove that there exists a subset D of the set of all labels such that for n  1, if r1r2 ...rn, s1s2 ... sn 2 D and r1r2 ¢ ¢ ¢ rn <l s1s2 ...sn, then wn[r1r2 ...rn] <l wn[s1s2 ...sn]. In other words, in the above tree, the nth-order α-words whose new labels belong to D are increasing from top to bottom in the lexicographic order.