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The th Dedekind number is the number of monotone Boolean functions of variables, for a nonnegative integer . Equivalently, it counts the antichains, including the empty one, in the power set of an -element set ordered by inclusion. For , 1, ..., is given by 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788, 286386577668298411128469151667598498812366, ... (OEIS A000372).

The numbers and their discoverers are summarized in the following table (Yusun 2011, Jäkel 2023). The last of these values was found by Jäkel (2023) using 5311 GPU-hours and 4257682565 matrix multiplications on Nvidia A100 GPUs.

The determination of these numbers is known as Dedekind's problem.

Pawelski et al. (2026) obtained the congruences

In particular, is odd. The computation counts fixed points of actions of finite -groups on monotone Boolean functions. These congruences do not determine the exact values of the three Dedekind numbers.

See also

Antichain, Boolean Function, Dedekind's Problem

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References

Church, R. "Numerical Analysis of Certain Free Distributive Structures." Duke Math. J. 6, 732-733, 1940.Church, R. "Enumeration by Rank of the Elements of the Free Distributive Lattice with Seven Generators." Not. Amer. Math. Soc. 12, 724, 1965.Dedekind, R. "Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Theiler." In Festschrift der Technischen Hochschule zu Braunschweig bei Gelegenheit der 69. Versammlung Deutscher Naturforscher und Ärzte. Braunschweig, Germany: Meyer, pp. 1-40, 1897. https://doi.org/10.1007/978-3-663-07224-9_1. Reprinted in Gesammelte mathematische Werke, Bd. 2 (Ed. R. Fricke, E. Noether, and Ø. Ore). Braunschweig, Germany: Vieweg, pp. 103-147, 1931. https://gdz.sub.uni-goettingen.de/id/PPN235693928.Jäkel, C. "A Computation of the Ninth Dedekind Number." J. Comput. Algebra 6-7, 100006, 2023. https://doi.org/10.1016/j.jaca.2023.100006.Sloane, N. J. A. Sequence A000372/M0817 in "The On-Line Encyclopedia of Integer Sequences."Pawelski, B.; Szepietowski, A.; and Szepietowski, B. "Congruences for Dedekind Numbers." 10 Nov 2026. https://arxiv.org/abs/2611.13887v1.Ward, M. "Note on the Order of the Free Distributive Lattice." Bull. Amer. Math. Soc. 52, 423, 1946.Wiedemann, D. "A Computation of the Eighth Dedekind Number." Order 8, 5-6, 1991.Yusun, T. J. L. "Dedekind Numbers and Related Sequences." Master's thesis. Burnaby, British Columbia, Canada: Simon Fraser University, 2011. https://summit.sfu.ca/item/12001.

Referenced on Wolfram|Alpha

Dedekind Number

Cite this as:

Weisstein, Eric W. "Dedekind Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DedekindNumber.html

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