Probability distributions of the numbers of configurations and discordances of random permutations from regular cyclic classes

The number p(s, s') of discordant positions, or discordances, of permutations s and s' acting on the set N-n = {1, 2,..., n} is rho(s, s') = \{i : s(i) not equal s'(i), i is an element of N-n}\. The position i is called non-discordant if s(i) = s'(i). Permutations s and s' of degree n are called discordant if rho(s, s') = n. Let xi(n)(l; d(l),..., d(m)) be the total number of non-discordant positions of a random permutation s from the cyclic class [l](n/l), consisting of n/1 cycles of length l, with fixed discordant permutations s(1),.., s(m), belonging to the cyclic classes s(i) is an element of [d(i)](n/di), 1 less than or equal to i less than or equal to m. The main results of the paper are the following two assertions. If m > 1 and l > 2 are fixed and n --> infinity, then the random variable xi(n)(l; d(1),..., d(m)) has in limit the Poisson distribution with parameter lambda = m. The random variable xi(n)(2; d(1),..., d(m)) has in limit, as n --> infinity, the distribution which is the composition of the Poisson distributions with parameters lambda(1) and lambda(2), where lambda(1) = 1/2 - gamma, lambda(2) = m - gamma, gamma = \{i : d(i) = 2, i = 1,..., m}\, namely, for any fixed r = 0, 1,..., P(xi(n)(2; d(1),..., d(m)) = r) --> e(-(m - y/2)) Sigma(v=0)([r/2]) (gamma/2)(v)(m-gamma)r-2v / v!(r-2v)!