  The partition function $p(n)$, which counts the number of partitions of a
positive integer $n$, is widely studied. Here, we study partition functions
$p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain
congruence conditions. A shifted partition identity is an identity of the form
$p_{S_1}(n-H) = p_{S_2}(n)$ for all $n$ in some arithmetic progression. Several
identities of this type have been discovered, including two infinite families
found by Alladi. In this paper, we use the theory of modular functions to
determine the necessary and sufficient conditions for such an identity to
exist. In addition, for two specific cases, we extend Alladi's theorem to other
arithmetic progressions.
