  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 $p(n)$.
