# Limiting distribution of generalized derangement

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.

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