Linear inequalities concerning the sum of the distinct parts congruent to r modulo m in all the partitions of n

Abstract

We investigate the sum of the parts congruent to r modulo m counted without multiplicity in all the partitions of n. Considering truncated  theta series, we introduce a collection of infinite families of linear inequalities involving this sum. We provide connections with  partitions with nonnegative rank, partition with non-negative crank and Garden of Eden partitions.