Bounds on the location of the maximum Stirling numbers of the second kind

Let S(n, k) denote the Stirling number of the second kind, and let K(n) be such that S(n, K(n) - 1) < S(n, K(n)) >= S(n, K(n) + 1). Using a probabilistic argument, we show that, for all n >= 2. left perpendiculare(w(n))right perpendicular - 2 <= K(n) <= left perpendiculare(w(n))right perpendicular + 1. where left perpendicularxright perpendicular denotes the integer part of x, and w(n) denotes Lambert's W function. (c) 2009 Elsevier B.V. All rights reserved.