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A counterexample to the triangle conjecture. (English) Zbl 0558.20032

It is known that the triangle conjecture sets a bound on the cardinality of a code formed by words of the form \(a^iba^j\). A counterexample exceeding this bound is presented in this paper. This also disproves a stronger conjecture that every code is commutatively equivalent to a prefix code.

MSC:

94A45 Prefix, length-variable, comma-free codes
05A99 Enumerative combinatorics
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References:

[1] de Felice, C., On the triangle conjecture, Inform. Process. Lett., 14, 197-200 (1982) · Zbl 0487.68074
[2] Hansel, G., Baïonnettes et cardinaux, Discrete Math., 39, 331-335 (1982) · Zbl 0486.20032
[3] Perrin, D.; Schützenberger, M. P., Un problème élémentaire de la théorie de l’information, (Théorie de l’Information. Théorie de l’Information, Colloq. Internat. CNRS No. 276 (1977)), 249-260 · Zbl 0483.94028
[4] Perrin, D.; Schützenberger, M. P., A conjecture on sets of differences of integer pairs, J. Combin. Theory Ser. B, 30, 91-93 (1981) · Zbl 0468.05033
[5] Pin, J. E.; Simon, I., A note on the triangle conjecture, J. Combin. Theory Ser. A, 32, 106-109 (1982) · Zbl 0472.05010
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