An Asymptotic Problem in Derangement Theory
Joseph Gillis, Mourad E. H. Ismail, T. Offer · SIAM Journal on Mathematical Analysis · 1990
N elements, divided into sets of respective cardinalities $\{ n_1 ,n_2 , \cdots ,n_a \} $, k sets of each size, where $N = k\sum_{i = 1}^a {n_i } $, are given. The probability is considered that a random permutation of the N elements is a derangement, i.e., that it leaves none of the elements in the set to which it belonged initially. In particular, an asymptotic estimate of this probability as $k \to \infty $ is obtained. It is known that the number of possible derangements can be represented by an integral involving products of Laguerre polynomials. The probability is obtained by asymptotic evaluation of a more general integral, involving the generalized Laguerre polynomials $L_n^{(\alpha )} (x)$, of which the integral required here is a special case.