Séminaire Lotharingien de Combinatoire, 93B.29 (2025), 11 pp.
Olivier Bernardi
Bijections for Faces of Braid-Type Arrangements
Abstract.
We establish a general bijective framework for encoding faces of some
classical hyperplane arrangements.
Precisely, we consider hyperplane arrangements in Rn whose
hyperplanes are all of the form
{xi-xj=s} for
some i,j ∈ [n]
and s ∈ Z.
Such an arrangement A is strongly transitive if it
satisfies the following condition: if
{xi-xj=s} ∉ A and
{xj-xk=t} ∉ A
for some i,j,k ∈ [n]
and s,t ∈ N,
then {xi-xk=s+t} ∉ A.
For any strongly transitive arrangement A, we establish a
bijection between the faces of A and some set of decorated plane
trees.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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