The Factor Spectrum and Derived Sequence 
Received:August 15, 2019 Revised:October 10, 2019 
Key Word:
kernel word envelope word return word derived sequence the factor spectrum

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11701024; 11431007). 

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Abstract: 
Given a sequence $\rho$ over a finite alphabet $\mathcal{A}$, an important topic in combinatorics on words is to find out all factors $\omega$ of $\rho$ and positive integers $p$ such that $\omega_p$ (the $p$th occurrence of $\omega$) fulfills property ${\mathcal{P}}$. This problem is equivalent to determining a notion called the factor spectrum. Determining the factor spectrum is a difficult problem. To this aim, we introduce several notions, such as: kernel word, envelope word, return word and derived sequence of each factor $\omega$. Using the factor spectrum and derived sequence, we can solve some enumerations of factors, such as the numbers of palindromes, fractional powers, etc. We will show some results for several sequences, such as the Fibonacci sequence, the Tribonacci sequence, the Perioddoubling sequence, etc. And we think that these notions and methods are suitable for all recurrent sequences. 
Citation: 
DOI:10.3770/j.issn:20952651.2019.06.015 
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