ON SEMIREGULAR PERMUTATIONS OF A FINITE SET

Abstract

In this paper we establish upper and lower bounds for the proportion of permutations in symmetric groups which power up to semiregular permutations (permutations all of whose cycles have the same length). Provided that an integer n has a divisor at most d, we show that the proportion of such elements in S-n is at least cn(-1+1/2d) for some constant c depending only on d whereas the proportion of semiregular elements in S-n is less than 2n(-1).