Young Diagrams in a Box — L(m,n) as a quotient of a Boolean algebra

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The callback to Chapter 5

Drag to pan, scroll or pinch to zoom. Hover an element to see its covers; click to pin it. The mn cells of an m×n box form a set, and its subsets under inclusion are the Boolean algebra B_mn. The group S_m ≀ S_n shuffles the cells within each row (a copy of S_n for each of the m rows) and shuffles the rows (S_m), so it has (n!)^m·m! elements. Shuffling within rows forgets which cells are chosen and keeps how many are chosen in each row; shuffling rows forgets which row had which count. What survives is the row counts sorted into decreasing order: a partition whose Young diagram fits in the box. Stanley's Theorem 6.9 says the orbit order (one orbit below another when some member of the first lies inside some member of the second) is containment of Young diagrams; the only real step is that if a_i ≤ b_i for every row, the jth largest a is at most the jth largest b. So L(m,n) is a quotient of a Boolean algebra, Chapter 5's theorem makes it rank-symmetric, rank-unimodal and Sperner, and the coefficients of the q-binomial coefficient, which are its rank sizes, are symmetric and unimodal. The orbits button slides every subset onto its Young diagram, so the cover edges of B_mn pile up on the cover edges of the quotient; each quotient edge is labeled with how many members of the upper orbit cover one fixed member of the lower orbit, the entries of the up operator on orbit sums. Within a row the diagrams are sorted by content (the sum of column minus row over their cells), so that conjugation is a left–right reflection when m = n.

Made for Math 372: Combinatorics at Reed College, fall 2026. L(m,n) is the interval below the m×n rectangle in Young's lattice (Stanley, Algebraic Combinatorics, Chapter 6, Theorem 6.9; Chapter 5 for quotients of Boolean algebras). Its rank-generating function is the q-binomial coefficient [m+n choose m]_q. Everything on this page is computed in the browser; the subsets are drawn only for mn ≤ 12, since B₁₆ has 65 536 elements, and the quotient stops at 7×7, which has 3432. Part of e-infinity.space/viz; see also the other Hasse diagrams.