Solving systems of equations on antichains for the computation of the ninth Dedekind Number

TL;DR

This paper introduces a method using P-coefficients to efficiently solve systems of equations on antichains, aiding the computation of the ninth Dedekind number D(9) and potentially improving calculations for larger n.

Contribution

It generalizes P-coefficients and demonstrates their application to multiple systems of equations, advancing techniques for computing Dedekind numbers.

Findings

Successfully computed D(9) using the proposed method

Generalized P-coefficients for broader applications

Potential to accelerate future Dedekind number calculations

Abstract

We study systems of equations on antichains, together with a way to count the number of solutions. We start with a simple example, generalise and show more applications. One of the results was used in the recent computation of D(9), the others have potential to speed up existing techniques in the future. In fact, the result of two independent computations of D(9) were published nearly at the same time, in one of them the authors of the present paper were involved. D(n) counts the monotone Boolean functions or antichains on subsets of a set of n elements. The number rises doubly exponentially in the number of elements n, and until now no algorithm of a lower combinatorial complexity is known to compute D(n). In our computation, we use coefficients representing the number of solutions of a specific set of equations on antichains over a finite set. We refer to these coefficients as P-coefficients. These can be computed efficiently. In this paper, we generalise this coefficient and apply it to four different systems of equations. Finally we show how the coefficient was used in our computation of D(9), and how its generalisations can be used to compute D(n).