Exact Dedekind numbers

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $[n]=\{1,\ldots,n\}$ and let $2^{[n]}$ be its power set. An antichain is a family $\mathcal A\subseteq2^{[n]}$ such that no distinct $A,B\in\mathcal A$ satisfy $A\subseteq B$. Define the Dedekind number \[ M(n)=\#\{\mathcal A\subseteq2^{[n]}:\mathcal A\text{ is an antichain}\}. \] The empty family and the family $\{\varnothing\}$ are both counted. Equivalently, $M(n)$ counts all maps $f:\{0,1\}^n\to\{0,1\}$ satisfying $f(x)\le f(y)$ whenever $x_i\le y_i$ for every $i$. The correspondence associates to $f$ the inclusion-minimal subsets on which $f$ has value $1$; thus both constant functions are included. Counting antichains removes the harmless choice between increasing and decreasing conventions for Boolean functions.

Statement recorded in the supplied source.

Determine the exact integer $M(n)$ for every $n\ge10$. The cited computation establishes $M(9)$ and develops exact counting formulae; an asymptotic estimate or a non-evaluated summation is not an exact value of the requested numbers. [S2]

Short English statement

Count exactly the antichains in the subsets of an $n$-element set, or equivalently all monotone Boolean functions, for every $n\ge10$.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3200013 (AMR-031-0013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Patrick De Causmaecker and Lennart Van Hirtum, Solving systems of equations on antichains for the computation of the ninth Dedekind Number (2024), §§1,2.1–2.2, pp.1–5. https://arxiv.org/abs/2405.20904.