Local probabilities for random permutations without long cycles

2014/12/31 by Manstavičius, Eugenijus, Petuchovas, Robertas
#05A15 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1501.00136

Abstract

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. 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. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.

Related

• The Cycle Structure of Permutations Without Long Cycles
• The number of cycles of specified normalized length in permutations
• The number of cycles in random permutations without long cycles is\n asymptotically Gaussian
• Recent results on permutations without short cycles
• Asymptotic analysis of the cyclic structure of permutations