2026/07/04 by Yanru Chen, Ang Li, Suijie Wang
#math.CO
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 ℓ satisfies 1≤ ℓ≤\lfloor m/2\rfloor, directed path-cover numbers when \lfloor m/2\rfloor<ℓ≤ m, and admissible gain-labeled arc sets for multilayer deletions. For ℓ=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.