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

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].