Let $\alpha\in(0;1)$ be an irrational number. Study of the
distribution of fractional parts $\{i\alpha\}$ on the interval
$(0;1)$ is a classical question in number theory. In particular,
H. Weyl proved that this sequence is uniformity distributed modulo 1. Since this work, various estimates for the remainder term of
the asymptotic formula for the number of the sequence of points
belonging to a given interval are actively investigated. Another
type of problems about considered sequence are problems associated
with the famous three lenghts theorem (Steinhaus conjecture),
which state that a tiling of the unit interval generated by the
points of the sequence, composed of intervals of two or three
different lengths. Moreover, in the second case the length of the
greatest inteval exactly equals the sum of the lengths of two
other intervals. It was find out that the geometry of these
tilings is closely connected with the first return maps for circle
rotations, Hecke–Kesten problem on bounded remainder sets,
combinatorics of Sturmian sequences, dynamics of two-color
rotations of the circle, and some other problems.
This paper deals to combinatorial properties of the sequence
$\{i\alpha\}$, such as permutations $\pi_{\alpha,n}$, generated by
the points $\{i\alpha\}$, $1\leq i\leq n$. It is proved that there
is a one-to-one correspondence between these permutations and the
intervals of Farey tilings of the level $n$. Here Farey tiling of
the level $n$ is a tiling of the interval $ [0;1]$ generated by
irreducible rational fractions of the form $ \frac{a}{b}$ with
denominator $0$. The proof is based on one theorem of
V. T. Sos, that allows to compute the permutation $\pi_{\alpha, n}$
using only $\pi_{\alpha,n}(1)$ and $\pi_{\alpha, n}(n)$. Also we
use the fact that the ends of intervals of the Farey coincide with
the points of discontinuity of the functions
$\{k\alpha\}-\{l\alpha\}$. As an application it is proved that
there are exactly $1+\sum_{k = 2}^n \varphi(k)$ different
permutations $\pi_{\alpha,n}$ for any fixed $n$. Another our
result states that the permutation $\pi_{\alpha,n}$ uniquely
determines permutations $\pi_{\alpha,m} $ with
$n$ and does not uniquely
determine the permutation $\pi_{\alpha,m} $ with
$m=\pi_{\alpha,n}(1)+\pi_{\alpha,n}(n)$.
