Rank partition functions and truncated theta identities

TL;DR

This paper introduces truncated theta identities related to partition ranks and cranks, deriving new inequalities for the partition function and exploring the concept of Garden of Eden partitions.

Contribution

It presents novel truncated forms of theta identities and establishes new infinite families of inequalities for the partition function p(n).

Findings

New truncated theta identities for partition ranks and cranks.

Infinite families of linear inequalities for p(n).

Analysis of Garden of Eden partitions related to p(n).

Abstract

In $1944$, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function $p(n)$. The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for $p(n)$.