×

zbMATH — the first resource for mathematics

Sur les facteurs des suites de Sturm. (On the factors of the Sturmian sequences.). (French) Zbl 0694.68048
Summary: We study a construction which connects to each line with slope p/q (such that \(\gcd (p,q)=1\) and \(q\leq n)\) a word of length n over the alphabet \(\{\) 0,1\(\}\). We show that this construction yields the language of all the factors of the Sturmian sequences. We first obtain a functional equation whose solution is the generating function of this language, and then we give a conjecture relating this generating function to the Euler function.

MSC:
68Q45 Formal languages and automata
05A15 Exact enumeration problems, generating functions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Autebert, J.M.; Beququier, J.; Boasson, L.; Nivat, M., Quelques problèmes ouverts en théorie des langages algébriques, RAIRO theoret. comput. sci., 13, 363-379, (1979) · Zbl 0434.68056
[2] Berstel, J., Transductions and context-free languages, (1979), Teubner Stuttgart · Zbl 0424.68040
[3] Berstel, J., Every iterated morphism yields a co-CFL, Inform. process. lett., 22, 7-9, (1986) · Zbl 0584.68082
[4] Christoffel, E.B., Observatio arithmetica, Ann. math., 6, 2, 148-152, (1875) · JFM 06.0113.02
[5] Cohen, H., Communication personelle, (1987)
[6] Coven, E.M.; Hedlund, G.A., Sequences with minimal block growth, Math. systems theory, 7, 138-153, (1983) · Zbl 0256.54028
[7] Flajolet, P., Analytic models and ambiguity of context-free languages, Theoret. comput. sci., 49, 283-309, (1987) · Zbl 0612.68069
[8] Grunbaum, B.; Shephard, G.C., Tilings and patterns, (1986), Freeman San Francisco · Zbl 0601.05001
[9] Hedlund, G.A., Sturmian minimal sets, Amer. J. math., 66, 605-620, (1944) · Zbl 0063.01982
[10] Hedlund, G.A.; Morse, M., Symbolic dynamics, Amer. J. math., 60, 815-866, (1938) · JFM 64.0798.04
[11] Hedlund, G.A.; Morse, M., Symbolic dynamics, part II: Sturmian trajectories, Amer. J. math., 62, 1-42, (1940) · JFM 66.0188.03
[12] W.F. Lunnon and P.A.B. Pleasants, Characterization of two distance sequences, Preprint. · Zbl 0759.11005
[13] Lunnon, W.F.; Pleasants, P.A.B., Quasicrystallographie tilings, J. math. pure appl., 66, 217-263, (1987) · Zbl 0626.52017
[14] Markoff, A.A., Sur une question de Jean Bernoulli, Math. ann., 19, 27-36, (1882) · JFM 13.0313.01
[15] Rauzy, G., Suites à termes dans un alphabet fini, Séminaire de théorie des nombres de Bordeaux, 25.01-25.16, (1982-1983)
[16] Rauzy, G., Mots infinis en arithmétique, dans: automata on infinite words, (), 165-171
[17] Series, C., The geometry of markoff numbers, Math. intelligencer, 7, 20-29, (1985) · Zbl 0566.10024
[18] Smith, H.J.S., Note on continued fractions, Messenger math., 6, 2, 1-14, (1876)
[19] Stolarsky, K.B., Beatty sequences, continued fractions and certain shift operators, Canad. math. bull., 19, 473-482, (1976) · Zbl 0359.10028
[20] Venkov, B.A., (), 67
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.