The number of cycles in random permutations without long cycles is asymptotically Gaussian

TL;DR

This paper proves that the total number of cycles in uniform random permutations without long cycles follows a Gaussian distribution asymptotically, and provides detailed asymptotic expansions for its mean and variance.

Contribution

It establishes a central limit theorem for permutations with restricted cycle lengths and derives asymptotic formulas for their statistical parameters.

Findings

Number of cycles is asymptotically Gaussian.

Asymptotic expansions for mean and variance are derived.

Results depend on the set of allowed cycle lengths.

Abstract

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we derive asymptotic expansions for the corresponding expected value and variance.