Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.4 |

Seilerwall 33

D-41747 Viersen

Germany

**Abstract:**

Let
denote the downset-lattice of the downset-lattice
of the finite poset *Q* and let *d*^{2}(*Q*) denote the cardinality of
.
We investigate relations between the numbers
*d*^{2}(*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
*d*^{2}(*Q*)^{3} < *d*^{2}(*A*_{1} + *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
*d*^{2}(*A*_{m} + *Q*) and
exponential sums of downset sizes and interval sizes related to
.
We apply these results in a comparison of the computational
times of algorithms for the calculation of the Dedekind numbers
*d*^{2}(*A*_{m}), including a new algorithm.

(Concerned with sequence A000372.)

Received January 20 2018; revised versions received May 1 2018; May 2 2018.
Published in *Journal of Integer Sequences*, May 8 2018.

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