# Algebraic Relations Between Partition Functions and the $j$-Function

## TL;DR

This paper establishes new identities linking the modular j-function with partition-related generating functions, using harmonic Maass forms and Hecke operators to express j-function coefficients combinatorially.

## Contribution

It introduces a novel formula for the Hecke action on a harmonic Maass form, connecting the j-function to partition statistics and unimodal sequences.

## Key findings

- Derived identities between the j-function and partition generating functions
- Expressed j-function coefficients in terms of combinatorial quantities
- Connected harmonic Maass forms with classical partition functions

## Abstract

We obtain identities and relationships between the modular $j$-function, the generating functions for the classical partition function and the Andrews $spt$-function, and two functions related to unimodal sequences and a new partition statistic we call the "signed triangular weight" of a partition. These results follow from the closed formula we obtain for the Hecke action on a distinguished harmonic Maass form $\mathscr{M}(\tau)$ defined by Bringmann in her work on the Andrews $spt$-function. This formula involves a sequence of polynomials in $j(\tau)$, through which we ultimately arrive at expressions for the coefficients of the $j$-function purely in terms of these combinatorial quantities.