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Suppose there are $N$ people in a party. Each of them brings $k$ gifts. When the party is over, each of them takes $k$ gift randomly. Denote $T$ is the number of gifts return to its original giver. Please find the limiting distribution with $N\to +\infty$ and $k$ fixed.

From this link Generalized Derangement Problem with $k$ unchanged elements , I find the generating function $$f(u)=\lim_{N\to \infty}\int_{0}^{+\infty}\frac{(k!)^N}{(Nk)!}(u-1)^{kN}L_k^N(\frac{x}{1-u})e^{-x}dx$$

However, I have trouble evaluate the coefficients for general $k$. I don't know much about the Laguerre polynomials.

Thanks in advance.

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  • $\begingroup$ Sorry, it has an answer here $\endgroup$ May 16, 2021 at 9:53
  • $\begingroup$ It turns out that I'm heading the wrong way. But I'll be grateful if anyone can evaluate the above integral. $\endgroup$ May 16, 2021 at 10:00

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