The number of symmetric chain decompositions

TL;DR

This paper derives asymptotic formulas for counting symmetric chain decompositions in Boolean lattices and hypergrids, revealing their exponential growth rates.

Contribution

It provides the first asymptotic enumeration formulas for symmetric chain decompositions in these combinatorial structures.

Findings

Number of symmetric chain decompositions of Boolean lattice is approximately (n/e)^{2^n}.

Number of symmetric chain decompositions of hypergrid is approximately n^{(1-o(1)) t^n}.

Establishes growth rates and asymptotic behavior of these decompositions.

Abstract

We prove that the number of symmetric chain decompositions of the Boolean lattice $2^{[n]}$ is $$\left(\frac{n}{2e}+o(n)\right)^{2^n}.$$ Furthermore, the number of symmetric chain decompositions of the hypergrid $[t]^n$ is $$n^{(1-o_n(1))\cdot t^n}.$$