  The dimension of an irreducible representation of $GL(n,\mathbb{C})$,
$Sp(2n)$, or $SO(n)$ is given by the respective hook-length and content
formulas for the corresponding partition. The first author, inspired by the
Nekrasov-Okounkov formula, conjectured combinatorial interpretations of
analogous expressions involving hook-lengths and symplectic/orthogonal
contents. We prove special cases of these conjectures. In the process, we show
that partitions of $n$ with all symplectic contents non-zero are equinumerous
with partitions of $n$ into distinct even parts. We also present Beck-type
companions to this identity. In this context, we give the parity of the number
of partitions into distinct parts with odd (respectively, even) rank. We study
the connection between the sum of hook-lengths and the sum of inversions in the
binary representation of a partition. In addition, we introduce a new partition
statistic, the $x$-ray list of a partition, and explore its connection with
distinct partitions as well as partitions maximally contained in a given
staircase partition.
