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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

Look at the following paper: Moni Naor, Omer Reingold: Constructing Pseudo-Random Permutations with a Prescribed Structure. J. Cryptology 15(2): 97-102 (2002) You can find it on Moni's site.
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