Relations between Powers of Dedekind
Numbers and Exponential Sums
Related to Them Frank a Campo
Seilerwall 33
D-41747 Viersen
Germany
mailto:acampo.frank@gmail.comacampo.frank@gmail.com

Abstract:

Let ${\cal D}^2 (Q)$ denote the downset-lattice of the downset-lattice of the finite poset Q and let d²(Q) denote the cardinality of ${\cal D}^2 (Q)$. We investigate relations between the numbers d²(A_m + Q)and their powers, where A_m is the antichain with m elements and A_m + Q the direct sum of A_m and Q. In particular, we prove the inequality d²(Q)³ < d²(A₁ + Q)² based on the construction of a one-to-one mapping between sets of homomorphisms. Furthermore, we derive equations and inequalities between the numbers d²(A_m + Q) and exponential sums of downset sizes and interval sizes related to ${\cal D}^2(A_m + Q)$. We apply these results in a comparison of the computational times of algorithms for the calculation of the Dedekind numbers d²(A_m), including a new algorithm.