Unlabeled simplicial complexes on n vertices ordered by subcomplex: the quotient J(Bn)/Sn
n
vertices
layout
orbit size 1n!
Made for Math 372: Combinatorics at Reed College, fall 2026. Elements are orbits of Sn on the simplicial complexes with vertex set [n] — the down-sets of the Boolean algebra Bn, including the void complex — ordered by inclusion; the order-reversing symmetry is Alexander duality. The unlabeled counts 3, 5, 10, 30, 210, 16 353 are the inequivalent monotone Boolean functions; n = 6 is the last one that can be drawn (there are 490 013 148 orbits for n = 7, and the labeled count for n = 9, the Dedekind number, was only found in 2023). The construction is the quotient of Chapter 5 of Stanley's Algebraic Combinatorics applied to the sub-lattice J(Bn) of B2n. Part of e-infinity.space/viz.