• Counting rises and levels in $r$-color compositions

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/127/02/0203-0217

• # Keywords

Combinatorial statistic; integer compositions; rise; level

• # Abstract

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.

• # Author Affiliations

1. Department of Mathematics, University of Haifa, 31905 Haifa, Israel

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019