The Bernardi Formula for Nontransitive Deformations of the Braid Arrangement

  • Ankit Bisain
  • Eric J. Hanson


Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum overĀ boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0 or $\pm 1$ 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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