2025/07/28 by P, Chaithra, Rani, Shushma, Venkatesh, R.
#05C15 #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2507.20847
We introduce the concepts of marked multi-colorings, marked chromatic polynomials, and marked (multivariate) independence series for hypergraphs. We show that the coefficients of the q-th power of the marked independence series of a hypergraph coincide with its marked chromatic polynomials in q, thereby generalizing a corresponding result for graphs established in Chaithra et al. 2025 (arXiv:2503.11230). These notions are then naturally extended to subspace arrangements. In particular, we prove that the number of marked multi q-colorings of a subspace arrangement is a polynomial in q. We also define the (marked) independence series for subspace arrangements and prove that the (-q)-th power of the independence series of a hyperplane arrangement has non-negative coefficients. We further conjecture that the (-q)-th power of the independence series of a hypergraph has non-negative coefficients if and only if all its edges have even cardinality.