Level Decompositions for Symmetric Deformations of the Braid Arrangement

2024/10/14 by Yanru Chen, Chen, Yanru, Suijie Wang +5 · 1 citation
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paper · pdf · doi:10.48550/arxiv.2410.10198

Abstract

Let A⊆\mathbb R_≥0 be finite and nonempty, and let \mathfrakA^A=(A₁^A,A₂^A,…) be the associated sequence of symmetric deformations of the braid arrangement. Denote by r_ℓ(A_n^A) the number of its level-ℓ regions and by F_ℓ(\mathfrakA^A,x) the corresponding exponential generating function. We prove F_ℓ(\mathfrakA^A,x)=(F₁(\mathfrakA^A,x))^ℓ. As a consequence, the characteristic polynomial has the binomial-basis expansion χ(A_n^A,t)=∑_ℓ=1ⁿ(-1)^n-ℓ r_ℓ(A_n^A)\binomtℓ. When 0∈ A and A^*=A∖\0\ is nonempty, we refine a classical identity of Stanley level by level: F_ℓ(\mathfrakA^{A^*},x)=F_ℓ(\mathfrakA^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 A_n^[0,m], we obtain r_ℓ(A_n^[0,m])=n! Ran_m+1,mℓ(n-ℓ), where 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.

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