Level of Faces for Exponential Sequence of Arrangements

TL;DR

This paper introduces a bivariate exponential generating function for level faces in exponential arrangements, revealing new combinatorial identities and extending classical polynomial formulas to Whitney polynomials.

Contribution

It establishes a new generating function formula for level faces, links face counts to Stirling numbers, and extends polynomial expansion theorems to Whitney polynomials.

Findings

Derived the formula $F_l(x,y)=(F_1(x,y))^l$ for level faces.

Proved an invariant sum of face counts equals Stirling numbers of the second kind.

Extended polynomial expansion theorems to Whitney polynomials.

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.