The number of binary rotation words
Anna E. Frid, Damien Jamet
TL;DR
This paper derives a precise formula and asymptotic behavior for counting binary rotation words generated by circle partitions, extending classical results on Sturmian words and their enumeration.
Contribution
It provides the first exact formula and asymptotic analysis for the number of binary rotation words of length n.
Findings
Number of binary rotation words of length n is asymptotically O(n^4)
Derived a precise enumeration formula for these words
Extended classical Sturmian word enumeration results
Abstract
We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be O(n^4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov in 1982, then independently by Berenstein, Kanal, Lavine and Olson in 1987, Mignosi in 1991, and then with another technique by Berstel and Pocchiola in 1993.
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