A noncrossing arc diagram is a collection of arcs joining pairs of vertically ordered points. Each arc descends monotonically from its upper endpoint to its lower endpoint and avoids the other marked points. Arcs can meet only at endpoints, and no two arcs have the same upper endpoint or the same lower endpoint. An arc is specified by its endpoints and the side on which it passes each intervening point (Reading 2015).
Reading (2015) gives a bijection between permutations of 1, ..., 
and noncrossing arc diagrams on 
points numbered from bottom to top. Each descent 
gives an arc from 
to 
, passing to the right
of intermediate values that occur before 
and to the left of intermediate values that occur after
.
The type-
Coxeter group consists of signed permutations
of the labels 
, ..., 
,
1, ..., 
satisfying 
. The map restricts to a bijection
between these signed permutations and the centrally
symmetric noncrossing arc diagrams on the 
labels (Barnard et al. 2026).
