Characteristic Polynomials of Graph- and Digraph-Deleted Catalan Arrangements

Abstract

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.

Disclosure

“age editing, computational checks, and the organization of the exposition. All mathematical results, proofs, and conclusions presented as contributions in this manuscript are the authors’ original work. The authors reviewed and revised all AI-assisted output and take full responsibility for the content of the manuscript. References [Abe24] T. Abe, T. N. Tran, and S. Tsujie, Vertex-weighted digraphs and freeness of arrangements between Shi and Ish, European Journal of Combinato”

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2026-08-24T00:28:51.963Z

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