Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.4

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

Mark Shattuck
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 are fixed and a ≥ 0 is arbitrary. Generating function and explicit formulas are found for general b and which extend earlier results when = 2 and b ≤ 3. We study the case = 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 as the part where is fixed.

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(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.

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