Algebraic Relations Between Partition Functions and the $j$Function
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.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 DOI:
 10.48550/arXiv.1907.07763
 arXiv:
 arXiv:1907.07763
 Bibcode:
 2019arXiv190707763L
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics
 EPrint:
 Res. number theory 6, 2 (2020)