Some New Restricted n-Color Composition Functions
Jarib R. Acosta, Yadira Caicedo, and Juan P. Poveda
Departamento de Matemáticas y Estadística
Universidad del Tolima
José L. Ramírez
Departamento de Matemáticas
Universidad Nacional de Colombia
Institute for Computational Science
Faculty of Mathematics and Statistics
Ton Duc Thang University
Ho Chi Minh City
An n-color composition is one in which a part of size m
can come in m colors (denoted by subscripts). Let C(ν)
denote the set of n-color compositions of the positive integer ν.
In this paper, we consider further modular restrictions on the subscripts
of the parts within members of C(ν). We first count
members of C(ν) in which all parts have subscripts of
the form ℓa+b, where b and ℓ are fixed
and a ≥ 0 is arbitrary. Generating function and explicit
formulas are found for general b and ℓ which extend
earlier results when ℓ = 2 and b ≤ 3. We study
the case ℓ = b-1 in further detail and find that the
corresponding subset of C(ν) is in bijection with various
classes of compositions. Finally, we consider two related problems:
one where the subscript restriction applies only to parts within a given
modular class and another where the subscript of a part belongs to the
same modular class mod ℓ as the part where ℓ is fixed.
Full version: pdf,
(Concerned with sequences
Received January 26 2019; revised version received August 15 2019.
Published in Journal of Integer Sequences,
August 24 2019.
Journal of Integer Sequences home page