Exact and limiting distributions of the number of successions in a random permutation

Abstract

Traditionally the distributions of the number of patterns and successions in a random permutation ofn integers 1,2, ..., andn were studied by combinatorial analysis. In this short article, a simple way based on finite Markov chain imbedding technique is used to obtain the exact distribution of successions on a permutation. This approach also gives a direct proof that the limiting distribution of successions is a Poisson distribution with parameter λ=1. Furthermore, a direct application of the main result, it also yields the waiting time distribution of a succession.