Sturmian words and the permutation that orders fractional parts

O'Bryant, Kevin

Journal of Algebraic Combinatorics, Tome 19 (2004) no. 1, pp. 91-115.

Résumé

Summary: A Sturmian word is a map $W : \mathbb N$ mathbbN $\mathbb N$ mathbbN 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 $pgr\_$ agr, n (where $agr$ is any irrational and $n$ is any positive integer) that orders the fractional parts $agr$, 2 $agr},\dots ,{ nagr}$, i.e., 0 $pgr\_{ $agr, n$ }(2)agr} pgr\_$ agr, n , and prove that for every irrational $agr$ there are infinitely many $n$ such that the order of $pgr\_{ $agr, n$ } (as an element of the symmetric group S \_{n})$ is less than $n$.

Détail

Keywords: Sturmian word, Beatty sequence, quasi crystal, sos permutation