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 , ⌊ e w ( n ) ⌋ − 2 ≤ K n ≤ ⌊ e w ( n ) ⌋ + 1 , where ⌊ x ⌋ denotes the integer part of x , and w ( n ) denotes Lambert’s W function.