Relations between Powers of Dedekind Numbers and Exponential Sums Related to Them

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.

Relations between Powers of Dedekind Numbers and Exponential Sums Related to Them

Frank a Campo Seilerwall 33 D-41747 Viersen Germany

Frank a Campo
Seilerwall 33
D-41747 Viersen
Germany