On the Number of Cycles in Randomly Selected Permutations

Abstract

Summary In this article we derive the distribution, mean and variance of the number of cycles in a randomly selected permutation of the first n integers.

Letter to the Editor

“…. , n } (Laghate and Deshpande 2003). This issue also contained my article giving the pmf of the number of surviving glacial moraines after n independent identically distributed (i.i.d.)…”

“…In case any of your readers missed it, the pmfs given in these two articles are identical! To see this, note that each article expresses the pmf through the same recurrence relation and boundary conditions (compare (2) of Laghate and Deshpande 2003 with (3) of Johnson 2003). So, we have a nice correspondence between the numbers of cycles, records, and surviving glacial moraines!…”

“…A couple of slight extensions to Laghate and Deshpande (2003) may then be made by using results for the equivalent problem in Johnson (2003). If X denotes the number of cycles in a randomly selected permutation of {1, 2, .…”

“…Also, the expression for the variance of X in Laghate and Deshpande (2003) necessarily simplifies to…”

“…95–9). The derivation of the mean and variance in Laghate and Deshpande (2003), however, is more direct than that of Knuth (1973), who uses a probability generating function approach.…”