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.

不可约表示的维数$GL（n,\mathbb{C}）$,$Sp（2n）$， 或者$S氧（n）$由相应分区的相应钩长度和含量公式给出。第一作者受到 Nekrasov-Okounkov 公式的启发，推测了涉及钩长度和辛/正交内容的类似表达式的组合解释。我们证明了这些猜想的特殊情况。在此过程中，我们证明所有辛内容非零的n的分区与n的分区是等数的分成不同的偶数部分。我们还为这一身份提供了贝克型同伴。在这种情况下，我们将分区数量的奇偶性赋予为具有奇数（分别为偶数）等级的不同部分。我们研究分区的二进制表示中钩长度总和与反转总和之间的联系。此外，我们引入了一种新的分区统计数据，即分区的X射线列表，并探索其与不同分区以及最大包含在给定楼梯分区中的分区的联系。