Permutations fixing a k - set

Let k, n be integers with 1 6 k 6 n/2 and select a permutation π ∈ Sn, that is to say a permutation of {1, . . . , n}, at random. What is i(n, k), the probability that π fixes some set of size k? Equivalently, what is the probability that the cycle decomposition of π contains disjoint cycles with lengths summing to k? Somewhat surprisingly, i(n, k) has only recently been at all well understood in the published literature. The lower bound limn→∞ i(n, k) ≫ log k/k is contained in a paper of Diaconis, Fulman and Guralnick [5], while the upper bound i(n, k) ≪ k may be found in work of Luczak and Pyber [9]. (These authors did not make any special effort to optimise the constant 1/100, but their method does not lead to a sharp bound.) Here and throughout X ≪ Y means X 6 CY for some constant C > 0. The notation X ≍ Y will be used to mean X ≪ Y and X ≫ Y . In the limit as n → ∞ with k fixed, a much better bound was very recently obtained by Pemantle, Peres, and Rivin [10, Theorem 1.7]. They prove that limn→∞ i(n, k) = k , where