| Summary: | 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.
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