  In this case-study in computer-human collaboration, we develop, implement,
and execute symbolic-computational algorithms for the automatic discovery and
proof of explicit expressions for the expectation, variance, and higher moments
of a large class of natural combinatorial statistics defined on Catalan-counted
objects, enabling, inter-alia, to prove that they are not asymptotically
normal. In particular, we reproduce in 0.12 seconds results of Miklos Bona, and
derive far deeper results, way beyond the scope of humans, concerning higher
moments of the random variable "number of occurrences of a pattern" in the set
of 132-avoiding permutations for all patterns of length 2 and 3, and, more
impressively, explicit expressions for the averages for all patterns of lengths
up to 10. The ample output inspired us to make an intriguing conjecture
concerning the number of so-called Bona classes, and we pledge to donate 100
dollars to the OEIS Foundation in honor of the prover (or disprover).
