In this paper, we investigate the factor properties and gap sequence of the Tri- bonacci sequence, the fixed point of the substitution σ（a, b, c） = （ab, ac, a）. Let Wp be the p-th occurrence of w and Gp（ω） be ...In this paper, we investigate the factor properties and gap sequence of the Tri- bonacci sequence, the fixed point of the substitution σ（a, b, c） = （ab, ac, a）. Let Wp be the p-th occurrence of w and Gp（ω） be the gap between Wp and Wp＋l. We introduce a notion of kernel for each factor w, and then give the decomposition of the factor w with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper： for each factor w, the gap sequence {Gp（ω）}p≥1 is the Tribonacci sequence over the alphabet {G1 （ω）, G2（ω）, G4（ω）}, and the expressions of gaps are determined completely. As an application, for each factor w and p C ∈N, we determine the position of Wp. Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.展开更多
Given a sequenceρover a finite alphabet A,an important topic in combinatorics on words is to find out all factorsωofρand positive integers p such thatωp(the p-th occurrence ofω)fulfills property P.This problem is...Given a sequenceρover a finite alphabet A,an important topic in combinatorics on words is to find out all factorsωofρand positive integers p such thatωp(the p-th occurrence ofω)fulfills property 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 factorω.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.展开更多
基金supported by grants from the National Science Foundation of China(114310071127122311371210)
文摘In this paper, we investigate the factor properties and gap sequence of the Tri- bonacci sequence, the fixed point of the substitution σ（a, b, c） = （ab, ac, a）. Let Wp be the p-th occurrence of w and Gp（ω） be the gap between Wp and Wp＋l. We introduce a notion of kernel for each factor w, and then give the decomposition of the factor w with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper： for each factor w, the gap sequence {Gp（ω）}p≥1 is the Tribonacci sequence over the alphabet {G1 （ω）, G2（ω）, G4（ω）}, and the expressions of gaps are determined completely. As an application, for each factor w and p C ∈N, we determine the position of Wp. Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1170102411431007)
文摘Given a sequenceρover a finite alphabet A,an important topic in combinatorics on words is to find out all factorsωofρand positive integers p such thatωp(the p-th occurrence ofω)fulfills property 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 factorω.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.
