Counting rises and levels in r-color compositions

An r-color composition of a positive integer n is a sequence of positive integers, called parts, summing to n in which each part of size r is assigned one of r pos- sible 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.