On the Average-Case Complexity of Selecting the k th Best

Let $\bar V_k (n)$ be the minimum average number of pairwise comparisons needed to find the $k$th largest of $n$ numbers $(k \geqq 2)$, assuming that all $n!$ orderings are equally likely. D. W. Matula proved that, for some absolute constant $c$, $\bar V_k (n) - n \leqq ck\ln \ln n$ as $n \to \infty $. In the present paper, we show that there exists an absolute constant $c' > 0$ such that $\bar V_k (n) - n \geqq c'k\ln \ln n$ as $n \to \infty $, proving a conjecture of Matula.