RANDOM PERMUTATIONS WITH PRIME LENGTHS OF CYCLES

A set of nth order permutations with prime lengths of cycles is considered. An asymptotic estimate for the number of all such permutations is obtained as n ->infinity. Given a uniform distribution on the set of such permutations of order n, a local limit theorem is proved, evaluating the distribution of the number of cycles v(n) in a permutation selected at random. This theorem implies, in particular, that the random variable v(n) is asymptotically normal with parameters (log log n, loglogn) as n ->infinity. It is shown that the random variable v(n)(p), the number of cycles of a fixed length p in such a permutation (p is a prime number), has in the limit a Poisson distribution with parameter 1/p. Assuming that a permutation of order n is selected in accordance with the uniform distribution from the set of all such permutations with prime cycle lengths, each of which has exactly N cycles (1 <= N <= [n/2]), limit theorems are proved, evaluating the distribution of the random variable mu(p)(n, N), the number of cycles of prime length p in this permutation. The results mentioned are established by means of the asymptotic law for the distribution of prime numbers and the saddle-point method as well as the generalized allocation scheme.