Local Probabilities for Random Permutations Without Long Cycles

  • Eugenijus Manstavičius
  • Robertas Petuchovas
Keywords: Symmetric group, Cycle structure, Short cycles, Saddle-point method

Abstract

We explore the probability $\nu(n,r)$ that a permutation sampled from the symmetric group of order $n!$ uniformly at random has no cycles of length exceeding $r$, where  $1\leq r\leq n$ and $n\to\infty$. Asymptotic formulas valid in specified regions for the ratio $n/r$ are obtained using the saddle-point method combined with ideas originated in analytic number theory.

Author Biographies

Eugenijus Manstavičius, Vilnius University
Professor at Vilnius University, Faculty of Mathematics and Informatics.
Robertas Petuchovas, Vilnius University
Doctoral student at Vilnius University, Faculty of Mathematics and Informatics.
Published
2016-03-18
Article Number
P1.58