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

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Kitaev, Sergey and Mansour, Toufik and Remmel, Jeffrey (2008) Counting descents, rises, and levels, with prescribed first element, in words. Discrete Mathematics and Theoretical Computer Science, 10 (3). ISSN 1365-8050

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Abstract

Recently, Kitaev and Remmel [9] re¯ned the well-known permutation statistic \descent" by ¯xing parity of one of the descent's numbers. Results in [9] were extended and generalized in several ways in [7, 10, 11, 12]. In this paper, we shall ¯x a set partition of the natural numbers N, (N1; : : : ;Ns), and we study the distribution of descents, levels, and rises according to whether the ¯rst letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = f1; : : : ; kg. In particular, we re¯ne and generalize some of the results in [4].

ORCID iDs

Kitaev, Sergey ORCID logoORCID: https://orcid.org/0000-0003-3324-1647, Mansour, Toufik and Remmel, Jeffrey;
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