## On the number of factors in codings of three interval exchange.(English)Zbl 1283.68274

Summary: We consider exchange of three intervals with permutation $$(3,2,1)$$. The aim of this paper is to count the cardinality of the set $$3\mathrm{iet}(N)$$ of all words of length $$N$$ which appear as factors in infinite words coding such transformations. We use the strong relation of $$3\mathrm{iet}$$ words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula $\#2\mathrm{iet}(N)/N^{3}\sim 1/\pi ^{2}$ for the number of Sturmian factors allows us to find bounds $1/3\pi ^{2} + o(1) \leq \# 3\mathrm{iet}(N)/N^{4} \leq 2/\pi ^{2} + o(1).$

### MSC:

 68R15 Combinatorics on words 05A05 Permutations, words, matrices 37B10 Symbolic dynamics

### Keywords:

interval exchange; enumeration of factors; Sturmian words
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