Distribution of Sample Arrangements for Runs Up and Down

Abstract

Using the notation of Levene and Wolfowitz [1], a new recursion formula is used to give the exact distribution of arrangements of $n$ numbers, no two alike, with runs up or down of length $p$ or more. These are tabled for $n$ and $p$ through $n = 14$. An exact solution is given for $p \geq n/2$. The average and variance determined by Levene and Wolfowitz are presented in a simplified form. The fraction of arrangements of $n$ numbers with runs of length $p$ or more are presented for the exact distributions, for the limiting Poisson Exponential, and for an extrapolation from the exact distributions. Agreement among the tables is discussed.