The Factor Spectrum and Derived Sequence

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 Period-doubling sequence, etc. And we think that these notions and methods are suitable for all recurrent sequences.