The Distribution of the Size of the Union of Cycles for Two Types of Random Permutations

We discuss some problems and permutation statistics involving two different types of random permutations. Under the usual model of random permutations, we prove that the shifted coverage of the elements of {1, 2, . . . , k} of a random permutation over {1, 2, . . . , n}; that is, the size of the union of the cycles containing these elements, excluding these elements themselves, follows a negative hypergeometric distribution. This fact gives a probabilistic model for the coverage via the canonical cycle representation. For a different random model, we determine some random permutation statistics regarding the problem of the lost boarding pass and its variations.