visualizations · posets
Hasse diagrams
Zoomable Hasse diagrams — unlabeled graphs, necklaces, simplicial complexes, finite spaces, the partition lattice, the Tamari lattice — drawn large enough to wander around in.
A Hasse diagram draws a finite poset by its cover relation: one dot per element, a segment from x up to y when y covers x, and nothing else — transitivity is left to the reader, and height reads as rank. Past a few dozen elements the drawing needs panning and zooming, which is what these maps are for.
Unlabeled graphs
BC(n,2)/Sn
Graphs on n ≤ 7 vertices up to isomorphism, ordered by “is isomorphic to a subgraph of.” The symmetric group acts on the Boolean algebra of edge sets, and the orbits are the graphs. Layouts by complement symmetry, degree spread, or |Aut|; a panel with automorphisms, degree sequence, covers, and the complement.
Necklaces
[k−1]n/Cn and [k−1]n/Dn
Necklaces or bracelets of n beads in 2, 3, or 4 colors, ordered bead by bead: a product of chains modulo the cyclic or dihedral group. With two colors this is exactly Stanley’s Bn/Cn. The panel gives the stabilizer, orbit size, color runs, covers, and the complementary necklace.
Simplicial complexes
J(Bn)/Sn
Simplicial complexes on n ≤ 6 vertices up to isomorphism, ordered by “is isomorphic to a subcomplex of”: the lattice of down-sets of Bn, modulo Sn. Rank is the number of faces, and the order-reversing symmetry is Alexander duality. The width of each quotient is computed for n ≤ 5, so you can check Sperner’s property by eye; n = 6 has 16 353 orbits, one per inequivalent monotone Boolean function.
Finite spaces
topologies on [n], modulo Sn
Topological spaces on n ≤ 6 points up to homeomorphism, ordered from coarser to finer: the poset of topologies on [n] modulo Sn. Each space is drawn as its open sets marked on Bn or as its specialization preorder; the mirror symmetry swaps open and closed sets. Rows count open sets, but the order is not graded — covers jump rows — so this is the one map where Sperner’s machinery does not apply.
Partition lattice
Πn and Πn/Sn, n ≤ 7
Set partitions of [n] under refinement, graded by the number of blocks, with the Möbius function and the product structure of every interval in the panel. A second view collapses the Sn-orbits to the integer partitions of n.
Tamari lattice
Tn+1 on trees and on transfer systems on [n]; NC(n+1)
One Catalan set drawn three ways: full binary trees with n+2 leaves, transfer systems on the chain [n], noncrossing partitions of [n+1]. The first two carry the Tamari order — a rotation is exactly adding relations — laid out as a planar projection of the 1-skeleton of Loday’s associahedron; the third carries Kreweras’ refinement order, coarser than Tamari, so toggling to it keeps every vertex in place and changes only the covers.
See also Sperner chains, a chain partition of Bn itself, and the Math 372 schedule, where these were built.