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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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:
  1. Narayana and Kirkman refinements;
  2. Reflection group generalizations;
  3. Fuss-Catalan and Rational Catalan generalizations (the problem session on type A Rational Catalan combinatorics was given by B. Rhoades and N. Williams); and
  4. $q$-Catalan and $(q,t)$-Catalan numbers as bigraded Hilbert series.
    1.     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.

      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 right-hand side as fibers of the Lyashko-Looijenga map.

          Cite this as: AimPL: Rational Catalan combinatorics, available at http://aimpl.org/rationalcatalan.