Level of Faces for Exponential Sequence of Arrangements
Combinatorics
2026-01-21 v1
Abstract
In this paper, we introduce the bivariate exponential generating function for the number of level- faces of an exponential sequence of arrangements (ESA), and establish the formula with a combinatorial interpretation. Its specialization at recovers a result first obtained by Chen et al. [3,4] for certain classic ESAs and later generalized to all ESAs by Southerland et al. [8]. As a byproduct, we obtain that an alternating sum of the number of level- faces is invariant with respect to the choice of ESA, and is exactly the Stirling number of the second kind. We also extend the binomial-basis expansion theorem [3,4,14] and Stanley's formula on ESAs [9] from characteristic polynomials to Whitney polynomials.
Keywords
Cite
@article{arxiv.2601.12328,
title = {Level of Faces for Exponential Sequence of Arrangements},
author = {Yanru Chen and Houshan Fu and Weikang Liang and Suijie Wang},
journal= {arXiv preprint arXiv:2601.12328},
year = {2026}
}