Combinatorial Applications of M\"obius Inversion

TL;DR

This paper uses Möbius inversion to establish combinatorial identities connecting various functions, extending previous work on partition functions and mock theta functions.

Contribution

It introduces a novel approach applying Möbius inversion to derive identities among combinatorial functions, moving beyond congruence relations.

Findings

Derived new combinatorial identities using Möbius inversion

Extended previous work on partition functions to a broader class of functions

Provided a framework for relating combinatorial functions via identities

Abstract

In important work on the parity of the partition function, Ono related values of the partition function to coefficients of a certain mock theta function modulo 2. In this paper, we use M\"obius inversion to give analogous results which relate several combinatorial functions via identities rather than congruences.