A Mean Paradox

Abstract

In a recent departmental research project, it was important to calculate certain statistics related to the average length of the cycles in a random permutation. By a well-known combinatorial result, when each of the permutations on n symbols is written as a product of disjoint cycles, the mean cycle length, averaged over all these cycles, is asymptotic (as n -> oo) to n/\ogn. However, when we did a computer simulation and averaged the cycle lengths over all the permutations that were generated, the mean cycle length was significantly larger than the n/ log n we were expecting. This is what we call the Mean Paradox.