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 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.
@article{10_37236_10233,
author = {Ankit Bisain and Eric Hanson},
title = {The {Bernardi} formula for nontransitive deformations of the braid arrangement},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10233},
zbl = {1476.05011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10233/}
}
TY - JOUR
AU - Ankit Bisain
AU - Eric Hanson
TI - The Bernardi formula for nontransitive deformations of the braid arrangement
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10233/
DO - 10.37236/10233
ID - 10_37236_10233
ER -
%0 Journal Article
%A Ankit Bisain
%A Eric Hanson
%T The Bernardi formula for nontransitive deformations of the braid arrangement
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10233/
%R 10.37236/10233
%F 10_37236_10233
Ankit Bisain; Eric Hanson. The Bernardi formula for nontransitive deformations of the braid arrangement. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10233
