This is actually about one particular question that I posted a while ago, "Special" meanders. Among several approaches tried is a huge subclass of approaches which can be generated from approaches to "usual" meanders. "Our" meanders seem to be essentially simpler - roughly, they relate to ordinary meanders in the same way as powers of two relate to Catalan numbers. So every time there is some approach in the general case, there is a hope it simplifies in our case.

So much for a motivation. Now one of these approaches is something called the method of matrix integrals. This method seems to have impressive applications in combinatorics. I've tried to find out how to apply this method in our case, and failed miserably. What is maddening is that I found several really excellent explositions - e. g. in the book Graphs on Surfaces by Lando and Zvonkin there is a very informative chapter about it. A survey about meanders by Michael La Croix also contains a separate chapter. Finally, there are several papers by Di Francesco and collaborators where this method is systematically used to obtain important estimates and conjectures about general meandric numbers.

And still when I start trying to apply this method to our particular problem, I am stuck from the very beginning. Maybe I am really stupid, I don't know - I simply cannot figure out how to start.

So the question is: does there exist in the literature some sort of cookbook for the matrix integral method? Something for dummies, some collection of algorithms, explicit steps, with an input in form of some combinatorial data?