We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous with the set of Motzkin paths. Such an equidistribution is established both analytically and bijectively.

翻译：我们证明了划分矩阵的“$\operatorname{inv}$”统计量的奇偶性有符号计数等于反演序列子类的基数。在建立这一结果的过程中，我们引入了一类有趣的划分矩阵，称为非正常划分矩阵。我们进一步证明，非正常划分矩阵的一个子集与Motzkin路径的集合等势。这种等分布性是通过解析和双射两种方式建立的。