论文标题
上限的属性对丰富的单词数量
Property of upper bounds on the number of rich words
论文作者
论文摘要
如果包含$ \ vert w \ vert+1 $不同的palindromic因子，包括空词，则称为\ emph {rich}，称为\ emph {rich}。令$ q \ geq 2 $为字母的大小。令$ r（n）$为长度$ n $的丰富单词的数量。令$ d> 1 $是一个真正的常数，让$ ϕ，ψ$是真实的函数，使得$ n_0 $有$ n_0 $，以至于所有$ n_0 $，以至于$ n_0 $，$ n_0 $ n> n_0 $，\ frac unpers $ \ f act $ \ frac and，长度的长度$ n $，\ item $ \ frac {x} {ψ（x）}+\ frac {x \ ln {（ϕ（x）}}} {ϕ（x）} $是一个严格增加的倒置功能。我们表明，如果$ C_1，C_2 $是真实常数，而$ r（n）\ leq q^{C_1 \ frac {n} {n} {ψ（n）}+c_2+c_2 \ frac {n \ frac {n \ ln（ϕ（n））} $ colution Austrant in suttort and suttor n stunte astor and instort and and instote a n s norte n and stante a n sacor an n n ostor $ n_0 $使得所有$ n> n_0 $我们都有\ [r（n）\ leq q^{（C_1+C_3）\ frac {n} {dψ（n）}+C_2 \ frac {n \ ln（ϕ（n））} {ϕ（n）}（1+ \ frac {1}
A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. Let $q\geq 2$ be the size of the alphabet. Let $R(n)$ be the number of rich words of length $n$. Let $d>1$ be a real constant and let $ϕ, ψ$ be real functions such that \begin{itemize}\item there is $n_0$ such that $2ψ(2^{-1}ϕ(n))\geq dψ(n)$ for all $n>n_0$, \item $\frac{n}{ϕ(n)}$ is an upper bound on the palindromic length of rich words of length $n$, and \item $\frac{x}{ψ(x)}+\frac{x\ln{(ϕ(x))}}{ϕ(x)}$ is a strictly increasing concave function. \end{itemize} We show that if $c_1,c_2$ are real constants and $R(n)\leq q^{c_1\frac{n}{ψ(n)}+c_2\frac{n\ln(ϕ(n))}{ϕ(n)}}$ then for every real constant $c_3>0$ there is a positive integer $n_0$ such that for all $n>n_0$ we have that \[R(n)\leq q^{(c_1+c_3)\frac{n}{dψ(n)}+c_2\frac{n\ln(ϕ(n))}{ϕ(n)}(1+\frac{1}{c_2\ln{q}}+c_3)}\mbox{.}\]
