Level of Faces for Exponential Sequence of Arrangements

TL;DR

This work analyzes level-$l$ faces in exponential sequences of arrangements (ESAs) through a bivariate exponential generating function $F_l(x,y)$, proving the fundamental factorization $F_l(x,y)=(F_1(x,y))^l$. It shows that the alternating sum over level-$l$ faces is invariant across ESAs and equals $l!\,S(n,l)$, tying the combinatorics to Stirling numbers of the second kind, and generalizes known results for characteristic polynomials to Whitney polynomials, extending binomial-basis expansions and Stanley-type identities. A bijective framework based on Hetyei’s labeling yields a concrete interpretation of the factorization, connecting level-$l$ faces to collections of level-1 faces across components. The results advance understanding of deformed braid ESAs and their Whitney polynomials, with parallel considerations for non-degenerate deformations of type-$B$ arrangements and potential broader combinatorial applications.

Abstract

In this paper, we introduce the bivariate exponential generating function $F_l(x,y)$ for the number of level-$l$ faces of an exponential sequence of arrangements (ESA), and establish the formula $F_l(x,y)=\big(F_1(x,y)\big)^l$ with a combinatorial interpretation. Its specialization at $x=0$ 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-$l$ 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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Zaslavsky2003, Theorem 3.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Theorem \ref{['stirling']}
• Lemma 2.3: Zhang2025, Theorem 1.1