Factorization of Fibonacci Words Involving Palindrome Inverses and Borders

• Main Authors: Yu-Ting Tseng, 曾毓婷
• Other Authors: Wai-Fong Chuan
• Format: Others
• Language: en_US
• Published: 2001
• Online Access: http://ndltd.ncl.edu.tw/handle/26557516375125133239

碩士 === 中原大學 === 數學研究所 === 89 === Abstract Let $A= { a,b }$. In [2], the $n$-th order Fibonacci word $w_n^{r_1r_2 ldots r_{n-2}}$ over $A$, where $r_1, cdots ,r_{n-2} in {0,1 }$, was defined recursively as follows: $w_1=a$, $w_2=b$, $w_3^0=ba$, $w_3^1=ab$, $w_4^{00}=bab$, $w_4^{11}=bab$, $w_4^{01}=bba$, $w_4^{10}=abb$, $w_k^{r_1 cdots r_{k-2}} = left { egin{array}{ll} w_{k-1}^{r_1r_2 cdots r_{k-3}}w_{k-2}^ {r_1r_2 cdots r_{k-4}}, & r_{k-2}=0 w_{k-2}^{r_1r_2 cdots r_{k-4}}w_{k-1}^ {r_1r_2 cdots r_{k-3}}, & r_{k-2}=1 end{array} ight.$ , for $5 leq k leq n$. Write $w_n^0$ for $w_n^{00 cdots 0}$, $n geq 4$. A nonempty word $x$ is called a $border$ of a nonempty word $w$ if there exist nonempty words $u$, $v$ such that $w=xu=vx$. A nonempty word $x$ is said to be a $left$ (resp., $right$) $palindrome$ $inverse$ of a nonpalindrome $w$, if $xw$ (resp., $wx$) is a palindrome and $|y| geq |x|$ whenever $y$ is a nonempty word such that $yw$ (resp., $wy$) is a palindrome. Here $|u|$ denotes the length of $u$. It was proved in [12] and [14] that the longest border of Fibonacci word $w_n^0$ is $w_{n-2}^0$. In this paper, we prove that borders of bordered Fibonacci words are themselves Fibonacci words. More precisely, the longest border of $w_n^{r_1r_2 ldots r_{n-2}}$ is $w_{n-2}^{r_1r_2 ldots r_{n-4}}$ if its label $r_1, ldots ,r_{n-2}$ ends with $00(10)^k$ or $11(01)^k$ for some $k$, $0 leq k leq frac{1}{2}(n-4)$; the longest border is $w_{n-1}^{r_1r_2 ldots r_{n-3}}$ if its label ends with $00(10)^{k-1}1$ or $11(01)^{k-1}0$ for some $k$, $1 leq k leq frac{1}{2}(n-3)$. The notion of left and right palindrome inverses was introduced by Lee in [15]. We proved that the products of the left and right palindrome inverses (also called LR-product and RL-product) of any nonpalindromic Fibonacci word are still Fibonacci words. Furthermore, the set of all the RL-products and LR-products of bordered, nonpalindromic Fibonacci words of order $n$ is precisely the set of all Fibonacci words of orders $n-1$ and $n-2$. Finally we obtain a factorization theorem of bordered, nonpalindromic Fibonacci words, involving their palindrome inverses and their longest borders.