2 1 Ja n 20 04 Derangements and asymptotics of Laplace transforms of polynomials

We describe the behavior as n → ∞ of the Laplace transforms of Pn, where P a fixed complex polynomial. As a consequence we obtain a new elementary proof of an result of Gillis-Ismail-Offer [2] in the combinatorial theory of derangements. 1 Statement of the main results The generalized derangement problem in combinatorics can be formulated as follows. Suppose X is a finite set and ∼ is an equivalence relation on X. For each x ∈ X we denote by x̂ the equivalence class of x. X̂∼ will denote the set of equivalence classes. The counting function of ∼ is the function ν = ν∼ : X̂ → Z, ν(x̂) = |x̂|. A ∼-derangement of x is a permutation φ : X → X such that x 6∈ x̂, ∀x ∈ X. We denote by N(X,∼) the number of ∼-derangements. The ratio p(X,∼) = N(X,∼) |X|! is the probability that a randomly chosen permutation of X is a derangement. In [1] S. Even and J. Gillis have described a beautiful relationship between these numbers and the Laguerre polynomials Ln(x) = e x d n dxn ( ex ) = n ∑ k=0 ( n k ) (−x)k k! , n = 0, 1, · · · . For example L0(x) = 1, L1(x) = 1− x, L2(x) = 1 2! (x − 4x+ 2).