1. Catalan Combinatorics and Reflection Groups
It is now generally recognized that there are two different kinds of Catalan objects: nonnesting and noncrossing. C. Stump gave the problem session on noncrossing objects from the point of view of the subword complex.V. Reiner summarized four directions of study for these objects:
 Narayana and Kirkman refinements;
 Reflection group generalizations;
 FussCatalan and Rational Catalan generalizations (the problem session on type A Rational Catalan combinatorics was given by B. Rhoades and N. Williams); and
 $q$Catalan and $(q,t)$Catalan numbers as bigraded Hilbert series.

A more complete introduction to this problem is given in V. Ripollâ€™s paper LyashokoLooijenga Morphisms and Submaximal Factorizations of a Coxeter Element [arXiv:1012.3825].
Let $W$ be a wellgenerated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \prod_{i=1}^n \frac{ph+d_i}{d_i}.$$
This is a suggestion for how to uniformly prove this formula for such $W$ (note that in the nonwellgenerated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.
We know casebycase that $Cat^p(W)$ counts multichains $w_1 \leq w_2 \leq \cdots \leq w_p \leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.
A $k$block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \cdots u_k,$$ such that $\sum_{i=1}^k \ell_T(u_i) = \ell_T(c) = n$ and $u_i \neq 1$.
Let $Fact_k(c)$ denote the number of $k$block factorizations of $c$.
Using the relation between multichains and strict chains, we obtain the identity $$\prod_{i=1}^n \frac{ph+d_i}{d_i} = \sum_{k=1}^n \binom{p+1}{k} Fact_k (c).$$
Despite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometricallyâ€”$Fact_k (c)$ counts the cardinality of fibers of the LyashkoLooijenga map. In particular, D. Bessis showed casebycase that the generic fibers of this map correspond to $n$block factorizations, or reduced decompositions in the reflections $T$ [arXiv:math/0610777]. Even though it was done casebycase, we have the uniform formula $$Fact_n(c) = \frac{n! h^n}{W} = \frac{(h)(2h)(3h)\cdots(nh)}{d_1 d_2 \cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.Problem 1.1.
[V. Ripoll] Uniformly prove $$\prod_{i=1}^n \frac{ph+d_i}{d_i} = \sum_{k=1}^n \binom{p+1}{k} Fact_k (c)$$ by interpreting the righthand side as fibers of the LyashkoLooijenga map.
Cite this as: AimPL: Rational Catalan combinatorics, available at http://aimpl.org/rationalcatalan.