Rational Catalan combinatorics

    A more complete introduction to this problem is given in V. Ripoll’s paper Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element [arXiv:1012.3825].

Let $W$ be a well-generated 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 non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.

We know case-by-case 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 geometrically—$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case 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 case-by-case, 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.