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.
Citation
P. S. Olmstead. "Distribution of Sample Arrangements for Runs Up and Down." Ann. Math. Statist. 17 (1) 24 - 33, March, 1946. https://doi.org/10.1214/aoms/1177731019
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