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arXiv · 2010.00930

The Bernardi formula for non-transitive deformations of the braid arrangement

Ankit Bisain·Eric J. Hanson
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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.

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Ankit Bisain, Eric J. Hanson. 2021-10-10. The Bernardi formula for non-transitive deformations of the braid arrangement. https://doi.org/10.37236/10233

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