Discrete Mathematics & Theoretical Computer Science, Vol 10, No 3 (2008)

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

Sergey Kitaev, Toufik Mansour, Jeff Remmel


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, (ℕ1, …, ℕs), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in ℕi 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.

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