Convolution numbers: the cyclic case

Abstract: Convolution sums are introduced and special instances of the cyclic convolution on finite sets is examined in more detail. The distributions that emerge are multidimensional generalizations of the Catalan and Narayana numbers. This work yields a closed form solution for 1-dimensional marginals and certain bivariate marginals in the cyclic prime case. It is explained how a sufficiently high resolution of understanding these multidimensional distributions yields an approach to attack the Hadamard matrix conjecture.