  An almost self-conjugate (ASC) partition has a Young diagram in which each
arm along the diagonal is exactly one box longer than its corresponding leg.
Classically, the ASC partitions and their conjugates appear in two of
Littlewood's symmetric function identities. These identities can be viewed as
Euler characteristics of BGG complexes of the trivial representation, for
classical Hermitian symmetric pairs. In this paper, we consider partitions in
which the arm-leg difference is an arbitrary constant $m$. Regarding these
partitions and their conjugates as highest weights, we prove an identity
yielding an infinite family of dimension equalities between $\mathfrak{gl}_n$-
and $\mathfrak{gl}_{n+m}$-modules. We then interpret this combinatorial result
in the context of blocks in parabolic category $\mathcal{O}$: using
Enright-Shelton reduction, we find six infinite families of congruent blocks
whose corresponding posets of highest weights consist of the partitions in
question. These posets, in turn, lead to generalizations of the Littlewood
identities and their corresponding BGG complexes. Our results in this paper
shed light on the surprising combinatorics underlying the work of Enright and
Willenbring (2004).
