Level Decompositions for Symmetric Deformations of the Braid Arrangement

Let $A\subseteq\mathbb R_{\ge0}$ be finite and nonempty, and let $\mathfrak{A}^A=(\mathcal{A}_1^A,\mathcal{A}_2^A,\ldots)$ be the associated sequence of symmetric deformations of the braid arrangement.

Denote by $r_\ell(\mathcal{A}_n^A)$ the number of its level-$\ell$ regions and by $F_\ell(\mathfrak{A}^A,x)$ the corresponding exponential generating function.

We prove $F_\ell(\mathfrak{A}^A,x)=\bigl(F_1(\mathfrak{A}^A,x)\bigr)^\ell$.

As a consequence, the characteristic polynomial has the binomial-basis expansion $\chi(\mathcal{A}_n^A,t)=\sum_{\ell=1}^{n}(-1)^{n-\ell}\,r_\ell(\mathcal{A}_n^A)\binom{t}{\ell}$.

When $0\in A$ and $A^*=A\setminus\{0\}$ is nonempty, we refine a classical identity of Stanley level by level: $F_\ell(\mathfrak{A}^{A^*},x)=F_\ell(\mathfrak{A}^{A},1-e^{-x})$.

Equivalently, the Catalan-type and semiorder-type level counts satisfy an unsigned Stirling convolution of the first kind.

For the $m$-Catalan arrangement $\mathcal{A}_n^{[0,m]}$, we obtain $r_\ell(\mathcal{A}_n^{[0,m]})=n!\,\operatorname{Ran}_{m+1,m\ell}(n-\ell)$, where $\operatorname{Ran}_{p,r}(q)$ is a Raney number.

This realizes Raney numbers as refined region counts and answers a question of Deshpande, Menon, and Sarkar.

The proofs use labeled Dyck paths, interval orders, and exponential sequences of arrangements.

We also realize the inverse Fu--Wang--Zhu bijection for $m$-Catalan regions by tableaux.