The Bernardi formula for non-transitive deformations of the braid arrangement
Abstract
Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0, +1, or -1 to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which are nested, recovering their known counting formula.
Keywords
Cite
@article{arxiv.2010.00930,
title = {The Bernardi formula for non-transitive deformations of the braid arrangement},
author = {Ankit Bisain and Eric J. Hanson},
journal= {arXiv preprint arXiv:2010.00930},
year = {2021}
}
Comments
23 pages, 9 figures. v3: journal version. v2: Level of generality changed in Sections 4-5 to correct an error in v1, additional clarity and examples added throughout