Integer compositions with part sizes not exceeding k
Abstract
We study the compositions of an integer n whose part sizes do not exceed a fixed integer k. We use the methods of analytic combinatorics to obtain precise asymptotic formulas for the number of such compositions, the total number of parts among all such compositions, the expected number of parts in such a composition, the total number of times a particular part size appears among all such compositions, and the expected multiplicity of a given part size in such a composition. Along the way we also obtain recurrences and generating functions for calculating several of these quantities. Our results also apply to questions about certain kinds of tilings and rhythm patterns.
Cite
@article{arxiv.1108.0337,
title = {Integer compositions with part sizes not exceeding k},
author = {Martin E. Malandro},
journal= {arXiv preprint arXiv:1108.0337},
year = {2012}
}
Comments
v2: title changed (previous title: Asymptotics for restricted integer compositions), paper rewritten with new focus. 16 pages, 4 tables