AI 中文总结
研究从完全 m-Catalan 排列通过删除由图、有向图和增益标记有向图索引的超平面得到的排列的特征多项式，给出了零层图删除的展开式及不同情况下系数的结果，还得到一些相关等式和应用。
AI 中文摘要
我们研究了通过删除由图、有向图和增益标记有向图索引的超平面从完全 m-Catalan 排列获得的排列的特征多项式。零层图删除给出了具有图形 Stirling 系数的降阶乘展开式。对于单个删除的平移层\(x_i - x_j = l\)，当\(1\leq l\leq\lfloor m/2\rfloor\)时系数是有向匹配数，当\(\lfloor m/2\rfloor < l\leq m\)时是有向路径覆盖数。对于多个平移层中的同时删除，系数是删除的增益标记弧的可允许集。在\(l = m\)的情况下，容斥展开式产生了一个将有向图及其补图的路径覆盖相关联的 Lah 数恒等式。我们还获得了整数线性分解的与 m 无关的准则、本质化后完全二分定向的紧凑区域计数公式以及有向 Ish 型应用。
英文摘要
We develop a finite-field stratification for characteristic polynomials of deletion subarrangements of the full $m$-Catalan arrangement. It reduces the count to cyclic placements of rigid blocks and yields falling-factorial expansions for deletions indexed by graphs, digraphs, and gain-labeled digraphs. The coefficients are graphical Stirling numbers for zero-layer deletions, directed matching numbers when the deleted layer $\ell$ satisfies $1\le \ell\le\lfloor m/2\rfloor$, directed path-cover numbers when $\lfloor m/2\rfloor<\ell\le m$, and admissible gain-labeled arc sets for multilayer deletions. For $\ell=m$, a complementary path-cover expansion yields factorization consequences. The method also gives formulas for directed Ish-type arrangements in terms of path-cycle covers and outdegrees.
Comments25 pages
