Random Permutations with Cycle Weights by Volker Betz1,

where (θ1, θ2, . . .)≡ θ are real nonnegative numbers, rj (π) denotes the number of j -cycles in π [we always have ∑ j jrj (π) = n] and hn is the normalization. We are mainly interested in the distribution of cycle lengths in the limit n→∞ and in how these lengths depend on the set of parameters θ . The probability P is really a probability on sequences r = (r1, r2, . . .) that satisfy ∑ j jrj = n. It is well known that r is the sequence of “occupation numbers” of a partition λ of n. That is, if λ denotes the partition λ1 ≥ λ2 ≥ · · · with ∑i λi = n, then rj is the number of λi that satisfies λi = j . Thus we are really dealing with random partitions. The number of permutations that are compatible with occupation numbers r is equal to n! ∏ j≥1 j rj rj ! .