Convolution numbers: the cyclic case

TL;DR

This paper explores cyclic convolution sums, revealing their connection to multidimensional Catalan and Narayana numbers, and provides solutions for certain marginals, offering insights potentially relevant to the Hadamard matrix conjecture.

Contribution

It introduces cyclic convolution sums, analyzes their distributions, and offers a closed form for specific marginals, advancing understanding of these mathematical structures.

Findings

Distributions generalize Catalan and Narayana numbers

Closed form solutions for 1D and some bivariate marginals

Potential approach to the Hadamard matrix conjecture

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.