Combinatorial interpretations of Ramanujan's tau function

TL;DR

This paper provides a combinatorial interpretation of Ramanujan's tau function using q-series identities, involving t-cores and a new class of partitions called (m,k)-capsids, offering alternative perspectives on its properties.

Contribution

It introduces a novel combinatorial framework for understanding the tau function through t-cores and (m,k)-capsids, expanding the interpretative tools for Ramanujan's identities.

Findings

Combinatorial interpretation of tau function via t-cores and (m,k)-capsids

Application of Ramanujan's q-series identity to partition theory

Potential for alternative interpretations using related identities

Abstract

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.