  We advertise elementary symmetric polynomials $e_i$ as the natural basis for
generating series $A_{g,n}$ of intersection numbers of genus g and n marked
points. Closed formulae for $A_{g,n}$ are known for genera $0$ and $1$ -- this
approach provides formulae for $g = 2,3,4$, together with an algorithm to
compute the formula for any g. The claimed naturality of the e_i basis relies
in the unexpected vanishing of some coefficients with a clear pattern: we
conjecture that $A_{g,n}$ can have at most $g$ factors $e_i$, with $i>1$, in
its expansion. This observation promotes a paradigm for more general cohomology
classes. As an application of the conjecture, we find new integral
representations of $A_{g,n}$, which recover expressions for the Weil-Petersson
volumes in terms of Bessel functions.
