  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 [Kiyoshi Igusa: Algebr. Geom. Topol. 4 (2004)
473-520]. The next coefficient was computed in [Kiyoshi Igusa: math.AT/0303157,
to appear in Topology]. 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 the last paper cited this verifies all of the formulas conjectured
by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705--749]. Mondello
[math.AT/0303207, to appear in IMRN] has obtained similar results using
different methods.
