Preserving the Number of Cycles of Length $k$ in a Growing Uniform Permutation

Abstract

The goal of this work is to describe a uniform generation tree for permutations which preserves the number of $k$-cycles between any permutation (except for a small unavoidable subset of optimal size) of the tree and its direct children. Moreover, the tree we describe has the property that if the number of $k$-cycles does not change during any $k$ consecutive levels, then any further random descent will always yield permutations with that same number of $k$-cycles. This specific additional property yields interesting applications for exact sampling. We describe a new random generation algorithm for permutations with a fixed number of $k$-cycles in $n+\mathcal{O}(1)$ expected calls to a random integer sampler. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter $1/k$.