Level Decompositions for Symmetric Deformations of the Braid Arrangement

Abstract

Let $A\subseteq {\mathbb{R}}_{\ge 0}$ be finite and nonempty, and let ${\mathfrak{A}}^A=({\mathcal{A}}_1^A,{\mathcal{A}}_2^A,\dots )$ 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 $F_{\ell }({\mathfrak{A}}^A,x)=(F_1({\mathfrak{A}}^A,x){)}^{\ell }$. As a consequence, the characteristic polynomial has the binomial-basis expansion $\chi ({\mathcal{A}}_n^A,t)={\sum }_{\ell =1}^n(-1{)}^{n-\ell }{r}_{\ell }({\mathcal{A}}_n^A)(\genfrac{}{}{0ex}{}{t}{\ell })$. When $0\in A$ and $A^{\ast }=A\setminus \{0\}$ is nonempty, we refine a classical identity of Stanley level by level: $F_{\ell }({\mathfrak{A}}^{A^{\ast }},x)=F_{\ell }({\mathfrak{A}}^A,1-e^{-x})$. 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 $r_{\ell }({\mathcal{A}}_n^{[0,m]})=n!{\mathrm{Ran}}_{m+1,m\ell }(n-\ell )$, where ${\mathrm{Ran}}_{p,r}(q)$ 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.

Substantial revisions: reorganized the structure of the paper, supplemented detailed reasoning for several proofs, fixed errors in arguments, improved exposition, and updated references