Subcube partitions of the vertex set of the n-cube: a two-slice recursion, the cubical Bell numbers, and the seventh term of A018926

The two-slice split identity for subcube partitions, for every n and every vertex set: pc X = sum over A subset of X₀ cap X₁ of pc A * pc (X₀ minus A) * pc (X₁ minus A); at the full cube, A018926(n+1) = sum over A subset V(Qₙ) of f(A)*f(Aᶜ)²; and its TWO-LEVEL form, A018926(n+2) = sum over pairs X₀, X₁ subset V(Qₙ) of two(X₀,X₁) * two(X₀ᶜ,X₁ᶜ)², via the equivalence Finset (Vtx (n+1)) Finset (Vtx n) x Finset (Vtx n) given by the two slices

A018926(0..4) = 1, 2, 8, 154, 89512 (the cubical Bell numbers Bᶜ₁..Bᶜ₅), each bound to the definition of a subcube partition rather than to a table

A018926(5) = 71319425714 (the sixth cubical Bell number Bᶜ₆) from the definition, via a table over the 65536 subsets of V(Q₄) that is checked entry-by-entry against the split identity

A018926(6) = 216009178641968077129336, the seventh cubical Bell number Bᶜ₇ – the number of ways to partition the vertex set of the 6-cube into subcubes

The number of CStrees on 7 binary variables is 7! * prod_(k=1..7) Bᶜₖ = 17125077428476440507901823199935254770103418880 (about 1.71251e46), the '?' at p = 7 in Duarte-Solus Figure 'Number of DAGs, CStrees and compatibly labeled staged trees'

Negative controls: A018926(3) is neither 153 nor 155 nor the Bell number 4140; the single free cube partitions Q₂ while the empty family does not and an overlapping pair does not; cell is injective at n = 2; the level-3 table check fails when one entry is corrupted; A018926(5) is neither of its neighbours; the certified table reproduces the independently proved A018926(4) at the full vertex set; the two-level identity reproduces A018926(3) = 154 at n = 1 and its summand is not constant; a Lean sweep chunk value is glued from five pieces by ordinary induction (sweepₐdd, sweepₐcc), not asserted

The companion data of arXiv:2101.09271 enumerates 61712 of the 89512 subcube partitions of Q₄: fourDpartitions.txt holds 61711 distinct valid records against threeDpartitions.txt's complete 153

A018926(6) recomputed inside Lean: a6chunk 0 65536 = 216009178641968077129336, the two-level split sum over bitmasks, evaluated by native_decide in five equal-cost chunks under –load-dynlib and glued by a kernel-clean range-splitting lemma

The companion sequence A018926 explicitly does not count: subcube partitions of Qₙ UP TO the hyperoctahedral group Bₙ (order 2ⁿ n!) are 1, 2, 4, 15, 434 for n = 0..4

The reach of a facet recursion on subcube partitions. For every n, every set T of directions, every direction i and every vertex set X, the partitions of X no block of which is free in a direction of T factor over the two halves at i: bnd (insert i T) X = bnd T (half i false X) * bnd T (half i true X); hence by inclusion–exclusion over the directions, the number of partitions of X in which every direction is spanned by a block is ∑_(T ⊆ Fin n) (-1)^|T| bnd T X. At the full cube: 9 of the 154 partitions of Q₃ and 18201 of the 89512 of Q₄ span every direction, so 145 and 71311 leave one free, and a facet recursion — the A = ∅ term of the split identity, with the one-block partition inserted by hand — tops out at 146 and 71312.

The 27800-partition shortfall of the companion 4-cube file of arXiv:2101.09271 decomposes exactly: 89512 = 61712 shipped + 9600 inside the recursion's reach but lost to a 146-element Q₃ seed + 18200 outside any facet recursion's reach.

Closed form for the facet-recursion blind spot: the number of subcube partitions of Qₙ in which every direction is spanned is ∑ₜ₌₀ⁿ (-1)ᵗ C(n,t) A018926(n-t)^(2ᵗ), giving 1, 1, 1, 9, 18201, 36714477073, 186447070849475267070945 for n = 0..6, so a facet recursion reaches 1, 2, 8, 146, 71312, 34604948642, 29562107792492810058392 — 94.8%, 79.7%, 48.5% and 13.7% of A018926(n) at n = 3, 4, 5, 6.