Increasing trees and Kontsevich cycles

Abstract

It is known that the combinatorial classes in the cohomology of the mapping class group of punctures surfaces defined by Witten and Kontsevich are polynomials in the adjusted Miller–Morita–Mumford classes. The first two coefficients were computed by the first author in earlier papers. The present paper gives a recursive formula for all of the coefficients. The main combinatorial tool is a generating function for a new statistic on the set of increasing trees on $2n+1$ vertices. As we already explained this verifies all of the formulas conjectured by Arbarello and Cornalba. Mondello has obtained similar results using different methods.