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Does anyone know a reference for the solution of the generalized derangement problem via Laguerre polynomials? The Wikipedia article here says that this is an application of inclusion-exclusion, but I don't see how.

This formula was used by joriki in a nice MSE answer here.

The article below solves the generalized derangement problem with inclusion-exclusion, but without Laguerre polynomials.

Reference: Finn F. Knudsen and Ivar Skau, "On the Asymptotic Solution of a Card-Matching Problem", Mathematics Magazine 69 (1996), 190-197.

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    @Mike Thanks for this, it looks like an interesting paper.2011-12-10

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The result is due to

J. Gillis and S.Even, "Derangements and Laguerre polynomials", Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79 / Issue 01, pp 135 - 143.

Link to the article: DOI:10.1017/S0305004100052154

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    The paper is placed in a wider context in [this paper](http://people.brandeis.edu/~gessel/homepage/papers/rookp.pdf) on rook polynomials.2011-12-27