Journal of Algebraic Combinatorics, 19 (1): 91--115, January 2004. MR2056768
A Sturmian word is a map W:N --> {0,1} for which the set of {0,1}-vectors
F_n(W):={ (W(i), W(i+1), ..., W(i+n-1) )^T : i \in N }
has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation p=p_{x,n} (where x is any irrational and n is any positive integer) that orders the fractional parts {x}, {2 x}, ..., {n x}, i.e.,
0 < {p(1) x} < {p(2) x} < ... < {p(n) x} < 1.
We give a formula for the sign of p_{x,n}, and prove that for every irrational x there are infinitely many n such that the order of p_{x,n} (as an element of the symmetric group S_n) is less than n.
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