Symmetric functions and a natural framework for combinatorial and number theoretic sequences

TL;DR

This paper explores a framework based on symmetric functions and power series triples that unifies and extends the understanding of various combinatorial and number theoretic sequences, revealing deeper structural insights.

Contribution

It extends previous work by Sun and Macdonald, providing a more comprehensive analysis of the structure of power series triples related to symmetric functions.

Findings

Unified framework for sequences like Stirling, Bernoulli, harmonic numbers, and partitions.

Identified identities linking power series triples through symmetric functions.

Highlighted the role of De Moivre polynomials in the structure of these sequences.

Abstract

Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power series in such a triple are closely linked by identities coming from the theory of symmetric functions. We extend the work of Z-H. Sun, who developed similar ideas, and Macdonald, revealing more of the structure of these triples. De Moivre polynomials play a key role in this study.