Increasing trees and Kontsevich cycles

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 leading coefficient was computed in [4]. The next coefficient was computed in [6]. 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 in [6] this verifies all of the formulas conjectured by Arbarello and Cornalba [1]. Mondello [10] has obtained similar results using different methods. AMS Classification numbers Primary: 55R40 Secondary: 05C05