# Laguerre polynomials and inclusion-exclusion

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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There's a related article in a recent issue of Mathematics Magazine that may interest you. See the Kayll reference in my blog post here. –  Mike Spivey Dec 9 '11 at 22:15
@Mike Thanks for this, it looks like an interesting paper. –  Byron Schmuland Dec 10 '11 at 1:20

## 1 Answer

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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Perfect! Thanks very much. –  Byron Schmuland Dec 9 '11 at 20:20
I added this as a reference in the Wikipedia article. –  joriki Dec 27 '11 at 15:47
The paper is placed in a wider context in this paper on rook polynomials. –  joriki Dec 27 '11 at 15:55