Averaging Sequences, Deranged Mappings, and a Problem of Lampert and Slater

We answer a question posed by D. E. Lampert and P. J. Slater (1998, Amer. Math. Monthly105, 556–558). Consider a sequence of real numbers qn in the interval [0, 1] defined by q0=0, q1=1, and, for n⩾1, qn+1 equals an average of preceding terms in the sequence. The weights used in the average are provided by a triangular array pn, k of probabilities whose row sums are 1. What is the limiting behavior of a sequence qn so defined? For the Lampert–Slater sequence the weight pn, k is the probability that a randomly chosen fixed-point free mapping of [n+1] omits exactly k elements from its image. To gain some insight into this averaging process, we first analyze what happens with a simpler array of weights pn, k defined in terms of binomial coefficients. One of our theorems states that if the weights pn, k are closely concentrated and the sequence qn exhibits oscillatory behavior up to a certain computable point, then it will exhibit oscillatory behavior from then on. We carry out the computations necessary to verify that the Lampert–Slater sequence satisfies the hypotheses of the latter theorem. A result on martingales (1992, N. Alon, J. H. Spencer, and P. Erdős, “The Probabilistic Method,” Wiley, New York) is used to prove the close concentration of the weights pn, k.