Convolution numbers

Function spaces defined on sets with a group action allow the definition of convolution integrals and sums. The simplest case involves binary functions on a finite set acted upon by a cyclic group. The convolution numbers that arise are multidimensional generalizations of the Catalan and Narayana numbers. I explain how a sufficiently detailed understanding of the multivariate distribution of such numbers, which I do not currently have, leads to a solution of several conjectures in number theory and combinatorics.