Characteristic Polynomials of Graph- and Digraph-Deleted Catalan Arrangements

We study characteristic polynomials of arrangements obtained from the full $m$-Catalan arrangement by deleting hyperplanes indexed by graphs, digraphs, and gain-labeled digraphs. Zero-layer graph deletions give falling-factorial expansions with graphical Stirling coefficients. For a single deleted translated layer $x_i-x_j=l$, the coefficients are directed matching numbers when $1\le l\le\lfloor m/2\rfloor$, and directed path-cover numbers when $\lfloor m/2\rfloor<l\le m$. For simultaneous deletions in several translated layers, the coefficients are admissible sets of deleted gain-labeled arcs. In the case $l=m$, an inclusion-exclusion expansion yields a Lah-number identity relating path covers of a digraph and its complement. We also obtain an $m$-independent criterion for integer linear factorization, compact region-count formulas for a complete bipartite orientation after essentialization, and directed Ish-type applications.