We study the row-space partition and the pivot partition on the matrix space ${\mathbb{F}}_q^{n\times m}$. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear rank-metric codes. We then generalize the Singleton-like bound for rank-metric codes, and introduce two new concepts of code extremality. Both of them generalize the notion of MRD code and are preserved by trace-duality. Moreover, codes that are extremal according to either notion satisfy strong rigidity properties analogous to those of MRD codes. As an application of our results to combinatorics, we give closed formulas for the $q$-rook polynomials associated with Ferrers diagram boards. Moreover, we exploit connections between matrices over finite fields and rook placements to prove that the number of matrices of rank $r$ over ${\mathbb{F}}_q$ supported on a Ferrers diagram is a polynomial in $q$, whose degree is strictly increasing in $r$. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions.

我们研究矩阵空间上的行空间分区和枢轴分区 ${\mathbb{F}}_q^{ñ\times 米}$。我们证明这两个分区都是自反的，行空间分区是自对偶的。此外，使用各种组合方法，我们显式计算与这些分区关联的Krawtchouk系数。这将为线性秩度量代码的行空间和枢轴枚举器建立MacWilliams类型的标识。然后，我们概括了秩度量代码的类Singleton界，并介绍了代码极端的两个新概念。两者都概括了MRD代码的概念，并通过跟踪对偶保留。此外，根据任一概念的极值代码都具有类似于MRD代码的强刚性特性。作为将结果应用于组合学的方法，我们给出了封闭式$q$与Ferrers图板关联的-rook多项式。此外，我们利用有限域上的矩阵与车流位置之间的联系来证明秩矩阵的数量$\mathrm{[R}$ 过度 ${\mathbb{F}}_q$ Ferrers图上支持的是多项式 $q$，其学位严格在 $\mathrm{[R}$。最后，我们研究了MacWilliams扩展定理的自然相似度，用于秩，行空间和枢轴分区。