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

Ankit Bisain, Eric J. Hanson

Published 2020-10-02Version 1

Bernardi has given a general formula to compute the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that the contribution to this sum of the set of boxed trees sharing an underlying rooted labeled tree is -1, 0, or 1 and give an algorithm for computing this value. We then restrict to arrangements which we call almost transitive and construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of rooted labeled trees in this case. We conclude by explicitly enumerating the trees corresponding to the regions of certain nested Ish arrangements which we call non-negative, recovering their known counting formula.