Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra: A brief note (pdf) on some relations among these topics, including the Catalan and Fuss-Catalan numbers. References are provided linking the analysis with the distribution of eigenvalues of random matrices, the Wigner semicircle law, and moduli spaces for marked Riemann surfaces.
For some notes on the Burgers equation (generalized form), see Lectures on Partial Differential Equations (Chapter 4) by R. Salmon.
See also the note on the inviscid Burgers equation in “Free probability theory: random matrices and von Nuemann algebras” by Voiculescu and page 112 of The Semicircle Law, Free Random Variables, and Entropy by Hiai and Petz.
(Added Sept. 2016):
See page 17 of “Mastering the master field” by Gopakumar and Gross (referenced in 2014 in the post “Appell polynomials, cumulants, …”
“On the large N limit of the Itzykson-Zuber Integral” by Matytsin