  We focus on combinatorial aspects of the Hilbert series of the cohomology
ring of the moduli space of stable pointed curves of genus zero. We show its
graded Hilbert series satisfies an integral operator identity. This is used to
give asymptotic behavior, and in some cases, exact values, of the coefficients
themselves. We then study the total dimension, that is, the sum of the
coefficients of the Hilbert series. Its asymptotic behavior involves the
Lambert W function, which has applications to classical tree enumeration,
signal processing and fluid mechanics.
