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Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:

a noncrossing matching on $2n$ vertices, and

a triangulation of an $n+2$-gon.

These objects are both counted by Catalan numbers, leading me to feel that there should be some connection. It is not the case that the fibres of the two maps are the same, and I will say more below about how they are clearly different. My question is: what is the connection?

Let me now explain the constructions. Consider the points $z$ in $\mathbb C$ such that arg($\pm f(z)$)$=\theta$. Generically, this consists of $n$ unbounded curves, which, for large $|z|$, are evenly-spaces spokes of a wheel (with $2n$ spokes), and therefore can be interpreted as giving rise to a noncrossing matching. See Martin, Savitt, Singer and Savitt. As you tune $\theta$, the matchings change in a nicely controlled way, as pairs of components meet and reconnect. Tuning from $\theta=0$ to $\theta=\pi$ results in $n-1$ of these reconnections, and the total effect is to rotate the diagram by one step.

Now we consider how to produce the triangulation. I found out about this from a sequence of papers by Cecotti, Vafa, and a varying list of others, studying a connection between BPS states of 4d supersymmetric quantum field theories and cluster algebras. For the present topic see, in particular, Sections 4 and 5 of Cecotti, Córdova, Vafa. I found background provided in Section 4 of Mulase and Penkava helpful. The idea is to define a foliation of $\mathbb C$, where a curve $p(t)$ lies on a leaf of the foliation if $(p'(t))^2f(p(t))e^{2\theta i}$ is always real and positive. From a zero of $f$, there will be three equally-spaced curves coming out, and generically they will connect to points at infinity. For $|z|$ very large, the trajectories approach spokes of a wheel, with $n+2$ spokes. By connecting into a triangle the endpoints (on the circle at infinity) of the three curves coming out from each zero of $f$, you get a triangulation of an $n+2$-gon. As you tune $\theta$, the triangulation changes by diagonal flips. Again, the total effect of tuning from 0 to $\pi$ is to rotate the diagram by one step (though note that the "step" is a different fraction of $2\pi$ from the step that appears in the noncrossing matching).

(Edited to add: I should mention that, in the terminology of the people who study these flows, $f(z)dz$ is called a "quadratic differential", though these are often considered in more complicated situations, where there might be poles, for instance.)

If you prefer to complicate matters rather than simplifying them, you could also consider the question of whether there are any other Catalan objects associated to a monic polynomial and a phase (with different fibres than the above maps).

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This reminds me of the article "On Stokes Sets", by Yuliy Baryshnikov [springerlink.com/content/lg2p8l0447n86075/], where Catalan numbers appear in a similar context. –  F. C. Feb 5 '12 at 13:18
    
Yes, absolutely. Thanks for pointing out the reference! For anyone who wants to get a better handle on the triangulation picture that I referred to, that's an excellent place to look. –  Hugh Thomas Feb 5 '12 at 14:13
    
About Baryshnikov's paper, I should add (because it confused me) that he is mainly interested in fixing theta, and letting the polynomial vary, whereas I asked about is fixing the polynomial and letting theta vary. (While in the paper by Cecotti et al, they actually vary both at once.) –  Hugh Thomas Feb 8 '12 at 12:44
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