A simple bijection for the regions of the Shi arrangement of hyperplanes

Noncrossing Partitions in Surprising Locations

“…They also show up in real hyperplane arrangements, Prüfer codes, quasisymmetric functions, and Hopf algebras, but detailing all of these connections would lead us too far afield. (see [1], [4], [5], [6], [8], [24], [46], [47], or [45] for details.) We conclude with one final illustration: the associahedron.…”

Sign Types Associated to Posets

“…The admissible sign types and some of its subfamilies were enumerated in all the cases (see [9,10,12]). Recently the sign types indexed by the root system of type A l have been studying extensively as hyperplane arrangements by quite a number of people (see [2,3,6,7,14,15]). …”

“…When I= [1,2,3], we arrange the entries of the upperunitriangular part of an I-sign type X in the following way.…”

“…By (1.4.1) and (1.4.2), we can reduce ourselves to the case of I= [1,2,3]. The latter can be checked directly from Example 1.2.…”

Hopf algebras and dendriform structures arising from parking functions

“…Geometrically, permutations correspond to chambers of the Coxeter arrangement of type A n−1 , while parking functions label those of the Shi arrangement [2]. But this is not the only possible explanation (see, e.g., [20]), and our choice was rather dictated by elementary combinatorial considerations (see Appendix).…”