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Reflection on the Reflection Complexity
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DOI：10.1007/s00224-026-10278-7.png)
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Abstract
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The factor complexity C-u of a sequence u = u(0)u(1)u(2)& centerdot;& centerdot;& centerdot; over a finite alphabet counts the number of factors of length n occurring in u, i.e., C-u(n) = #L-n(u), where L-n(u) = {u(i) u(i)+1 & centerdot;& centerdot;& centerdot; u(i+n-1): i is an element of N}. Two factors of L-n(u) are said to be equivalent if they are equal or one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity ru which counts the number of non-equivalent factors of L-n(u). They formulated the following conjecture: a sequence u is eventually periodic if and only if r(u)(n + 2) = ru(n) for some n is an element of N. Here we prove the conjecture and characterize the sequences for which r(u)(n + 2) = ru(n) + 1 for every n is an element of N and also the sequences for which the equality is satisfied for every sufficiently large n is an element of N.
Keywords:
Reflection complexity
Quasi-Sturmian sequences
Factor complexity
Periodicity
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0.4
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