Rank partition functions and truncated theta identities

Mircea Merca

doi:10.2298/aadm190401023m

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).

Highlights

A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n [1]
Theorem 1 has opened up a new study on truncated theta series and linear partition inequalities
Other recent investigations involving truncated theta series and linear partition inequalities can be found in several papers by Andrews and Merca [5], Chan, Ho and Mao [10], Guo and Zeng [14], He, Ji and Zang [15], Mao [17, 18], Merca [19, 20, 21], and Merca, Wang and Yee [22]

Summary

INTRODUCTION

A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n [1]. As a consequence of Theorem 1, Andrews and Merca derived the following linear partition inequality: For n > 0, k 1, k−1. Theorem 1 has opened up a new study on truncated theta series and linear partition inequalities. Other recent investigations involving truncated theta series and linear partition inequalities can be found in several papers by Andrews and Merca [5], Chan, Ho and Mao [10], Guo and Zeng [14], He, Ji and Zang [15], Mao [17, 18], Merca [19, 20, 21], and Merca, Wang and Yee [22]. An immediate consequence owing to the positivity of the sums on the right hand side of the second identity is given by the following infinite family of linear partition inequalities. Connections between partitions with rank −2 or less and partitions with positive crank are given in this context

PROOF OF THEOREM 2

TRUNCATED IDENTITY OF AULUCK

GARDEN OF EDEN PARTITIONS

CONCLUDING REMARKS