##
**
Some New Restricted ***n*-Color Composition Functions

###
Jarib R. Acosta, Yadira Caicedo, and Juan P. Poveda

Departamento de Matemáticas y Estadística

Universidad del Tolima

Tolima

Colombia

José L. Ramírez

Departamento de Matemáticas

Universidad Nacional de Colombia

Bogotá

Colombia

Mark Shattuck

Institute for Computational Science

and

Faculty of Mathematics and Statistics

Ton Duc Thang University

Ho Chi Minh City

**Abstract:**

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,
dvi,
ps,
latex
**

(Concerned with sequences
A000045
A000930
A003269
A003520
A005708
A052908
A116732
A176848.)

Received January 26 2019; revised version received August 15 2019.
Published in *Journal of Integer Sequences*,
August 24 2019.

Return to
**Journal of Integer Sequences home page**