Peaks and Eulerian numbers in a random sequence
Di Warren, Eugene B. Seneta · Journal of Applied Probability · 1996
We consider the exact distribution of the number ofpeaksin a random permutation of the integers 1, 2, ···,n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem ofascents(which has been thoroughly analysed) and both our problem ofpeaksand similar problems on thecircle. This link then permits easy deduction of certain general properties, such as linearity innof the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting ofpeaks and troughs.