Sturmian Words and the Permutation that Orders Fractional Parts
| dc.creator | O'Bryant, Kevin | |
| dc.date | 2002-11-13 | |
| dc.date.accessioned | 2026-07-07T04:52:52Z | |
| dc.date.available | 2026-07-07T04:52:52Z | |
| dc.description | A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} 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 $π$ (depending on an irrational x and a positive integer n) that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {π(1) x} < {π(2) x} < ... < {π(n) x} < 1. We give a formula for the sign of $π$, and prove that for every irrational x there are infinitely many n such that the order of $π$ (as an element of the symmetric group S_n) is less than n. | |
| dc.description | 20 pages, 1 figure, Mathematica notebook available from author | |
| dc.identifier | https://arxiv.org/abs/math/0211200 | |
| dc.identifier | http://arxiv.org/abs/math/0211200 | |
| dc.identifier | J. Algebraic Combin. 19 (2004), no. 1, 91--115. | |
| dc.identifier | doi:10.1023/B:JACO.0000022568.96268.12 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65634 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | 68R15; 11J71; 05E99 | |
| dc.title | Sturmian Words and the Permutation that Orders Fractional Parts | |
| dc.type | text |