Colored base-3 partitions, sequences of polynomials, and perfect numbers

2025/09/03 by Karl Dilcher, Dilcher, Karl, Larry Ericksen +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications

paper · pdf · doi:10.48550/arxiv.2509.03147

Abstract

Motivated by the observation that the counting function of a certain base-3 colored partition contains the even perfect numbers as a subsequence, we begin by defining a sequence of polynomials in four variables and discuss their properties and combinatorial interpretations. We then concentrate on certain subsequences that are related to the Chebyshev polynomials of both kinds. Finally, we consider several sequences of single-variable polynomials that have meaningful combinatorial interpretations as well as interesting zero distributions.

Citations

• Polynomials and algebraic curves related to certain binary and b-ary overpartitions
• Polynomial analogues of restricted multicolor b-ary partition functions
• On p-adic valuations of colored p-ary partitions
• Restricted k-color partitions
• Combinatorial results for semigroups of order-preserving partial transformations

Related

• Counting with two-level polynomials
• Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros
• Polynomial analogues of restricted multicolor b-ary partition functions
• A nonstandard technique in combinatorial number theory
• Two-color partitions with evens in one color