Counting descents, rises, and levels, with prescribed first element, in words

Abstract

Recently, Kitaev and Remmel refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers ℕ, (ℕ 1, …, ℕ s), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in ℕ i over the set of words over the alphabet [k]= {1,…,k}. In particular, we refine and generalize some of the results by Burstein and Mansour.