An $r$-color composition of a positive integer $n$ is a sequence of positiveintegers, called parts, summing to n in which each part of size $r$ is assigned one of $r$ possible colors. In this paper, we address the problem of counting the $r$-color compositions having a prescribed number of rises. Formulas for the relevant generating functions are computed which count the compositions in question according to a certain statistic. Furthermore, we find explicit formulas for the total number of rises within all of the $r$-color compositions of $n$ having a fixed number of parts. A similar treatment is given for the problem of counting the number of levels and a further generalization in terms of rises of a particular type is discussed.
Volume 130, 2020
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