Distribution of Number of Levels in an $$[\varvec{s}]$$-Specified Random Permutation

Abstract

Successions, Eulerian and Simon Newcomb numbers, and levels are the best-known patterns associated with \([\varvec{s}]\)-specified random permutations. The distribution of the number of rises was first studied in 1755 by Euler. However, the distribution of the number of levels in an \([\varvec{s}]\)-specified random permutation remained unknown. In this study, our main goal is to identify the distribution of the number of levels, which we achieve using the finite Markov chain imbedding technique and insertion procedure. An example is given to illustrate the theoretical result.