The proportion of permutations fixing a $k$-set

TL;DR

This paper derives an asymptotic formula for the probability that a large permutation has an invariant subset of size k, involving a smooth function and a power-logarithmic decay.

Contribution

It provides the first explicit asymptotic formula for the probability of fixing a k-set in permutations, including the behavior of an associated oscillating function.

Findings

Asymptotic formula for p(k) involving a smooth function f.

The function f varies very little, with a ratio less than 1 + 2×10^{-7}.

Conjecture that f is not constant, indicating subtle oscillations.

Abstract

Denote by $p(k)$ the limit, as $n \rightarrow \infty$, of the probability that a random permutation on a set of size $n$ has an invariant set of size $k$. We give an asymptotic formula for $p(k)$, showing that it is asymptotically $f(\{\log_2 k\}) k^{-\delta} (\log k)^{-3/2}$ where $\delta = 1 - \frac{1 + \log \log 2}{\log 2} \approx 0.086$ and $f$ is a smooth, positive, function on $\mathbb{R}/\mathbb{Z}$, which we will describe explicitly. The function $f$ satisfies $\frac{\max f}{\min f} < 1 + 2 \times 10^{-7}$ and we conjecture that it is not constant. Estimating $p(k)$ is a model for the more well-known question which asks for an estimation of $M(n)$, the number of distinct elements in the $n$-by-$n$ multiplication table. By elaborating on the techniques in this paper, we will give an asymptotic for $M(n)$ in forthcoming work.