# Path-type description for (faces of) associahedra?

Recall that faces of associahedra are indexed by planar trees aka configurations of non-interesecting diagonals in polygons. And incidence corresponds to contracting edges / removing diagonals.

Vertices of associahedra can also be indexed by Dyck paths. What is the corresponding interpretation for all faces? How can incidence be described in these terms?

Some thoughts. Actually, I'm quite sure that faces are indexed by (small) Schröder paths.

Heuristic explanation: edges correspond to “elementary switches” and the most natural transformation one can do to a Dyck path is to switch RU<->UR (R standing for a step right, U — for a step up, naturally); let's mark the place of switches by diagonal steps — now we get Schröder paths (small Schröder paths, actually: diagonal steps can't lie on the main diagonal — or the “UR end” of the switch would intersect the main diagonal). Note that this heuristic also describes the incidence relation — but this description is clearly wrong: it predicts that the vertex $R^nU^n$ lies only on one edge (namely, on the $U^{n-1}DR^{n-1}$).

Recall that bijection between non-associative products and Dyck paths is given by Reverse Polish notation — so swtiches should correspond not to corners but just to vertical steps (and the symmetry between horizontal and vertical steps breaks). Anyway, now it's not hard to write down some examples. For 2-associahedron aka pentagon, say: the cyclic order of vertices coming from $$((ab)c)d\to (a(bc))d\to a((bc)d)\to a(b(cd))\to (ab)(cd)\to$$ is $$RURURU\to RRUURU\to RRURUU\to RRRUUU\stackrel{(!)}\to RURRUU\to$$ (note the step (!) not agreeing with naive heuristic).

But I have no idea how to describe the incidence relation in general case.

-
 I don't know if this helps, but browsing a little revealed these nice notes by Fomin and Reading, which describe the $(n-3)$-dimensional associahedron in terms of triangulations of the $n$-gon. Very nice. This doesn't quite yield a description via Dyck paths, but on the other hand why insist? After all, there are many things enumerated by the Catalan numbers and maybe Dyck paths are not exactly the handiest implementation? – t.b. Aug 15 '11 at 9:31 @Theo Yes, this description (which is, up to obvious bijection, coincides with the original definition, I believe) is definitely very clear. The motivation for the question (well, beyond natural curiosity) is that there is some embedding of n-associahedron into Euclidean space that is easier to describe in terms of paths. – Grigory M Aug 15 '11 at 9:37