Partition Lattice — set partitions of [n] under refinement

Hover an element

Drag to pan, scroll to zoom. Hover an element to see its covers; click to pin it. The elements of Π_n are the set partitions of [n] = {1, …, n}, ordered by refinement: π ≤ σ when every block of π lies inside a block of σ. Going up one step merges two blocks, so the lattice is graded by n minus the number of blocks, with S(n, k) elements in the row with k blocks and B(n) elements in all. Each glyph places 1, …, n clockwise around a circle from the top and draws every block of size two or more as a chord or a filled polygon. Within a row the partitions are in canonical order — by the block of 1, then the block of 2, and so on. Since restricting a partition of [n] to [m] preserves refinement, each row is sorted into nested clusters of partitions with the same restriction to [m], and a cover either stays inside its cluster or moves between clusters the way the corresponding cover of Π_m does; copies of the smaller lattices are visible inside the larger. The symmetric group permutes the ground set and preserves refinement; the orbits button draws the quotient poset Π_n/S_n, whose elements are the integer partitions of n (the multiset of block sizes), one orbit below another when some partition of the first type refines some partition of the second. Color is the number of blocks, or one hue per orbit; switching to the orbits view first colors every partition by its orbit, then slides each orbit together onto its point of the quotient, so that the cover edges of Π_n pile up on the cover edges of the quotient — the quotient edges are drawn with a width that records how many ways a representative of the lower orbit is covered inside the upper one.

Made for Math 372: Combinatorics at Reed College, fall 2026. Π_n is the lattice of set partitions of [n] under refinement (Stanley, Enumerative Combinatorics I, §3.10; Algebraic Combinatorics, Chapter 5 for the quotient by S_n). It is graded by n minus the number of blocks, its rank sizes are the Stirling numbers of the second kind, every interval is a product of smaller partition lattices, and μ(0̂, 1̂) = (−1)ⁿ⁻¹(n−1)!. Everything on this page is computed in the browser; n stops at 7 because B(8) = 4140 elements no longer fit in a picture. Part of e-infinity.space/viz.