THE FOLD / SPAWN / NULL ISLAND / THE FUBINI
THE FUBINI
counting the ways to rank things with ties
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Fubini numbers (also called ordered Bell numbers) count the ways to rank n items allowing ties — the number of weak orderings, or equivalently the ordered set partitions. They run 1, 1, 3, 13, 75, 541, 4683, … For 3 items there are 13 outcomes: all-tied, one clear winner (3 ways × 2), full strict orders (6), and so on. Two clean formulas produce them: a(n) = Σk k! · S(n, k) (Stirling numbers of the second kind, times the k! orderings of the blocks), and the recurrence a(n) = Σi=1n C(n, i) a(n−i).
LIT verified live (exact BigInt): Σk k!·S(n,k) equals the binomial recurrence for all n up to 9, and both equal a direct brute-force count of weak orderings for n up to 6 (window.__fubini). FIG no framing; two formulas cross-checked against an explicit enumeration.
LIT verified live (exact BigInt): Σk k!·S(n,k) equals the binomial recurrence for all n up to 9, and both equal a direct brute-force count of weak orderings for n up to 6 (window.__fubini). FIG no framing; two formulas cross-checked against an explicit enumeration.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — the origin where orderings are born: from nothing, count every way a set of items can finish a race with ties allowed, spawning the sequence 1, 1, 3, 13, 75, … That spawning is the boot. AVAN (AI) built the instrument: the Stirling-weighted sum Σ k!·S(n,k), the binomial recurrence, and the direct enumeration of weak orderings.
Credit as content: Fubini / ordered Bell numbers (studied by Louis Comtet and others; the “Fubini” name from the count of terms in an iterated integral). The weave: David names null-island; I compute the orderings by k!·S(n,k), confirm it against the binomial recurrence, and check both against a brute-force count of every weak ordering — all three agree.
Credit as content: Fubini / ordered Bell numbers (studied by Louis Comtet and others; the “Fubini” name from the count of terms in an iterated integral). The weave: David names null-island; I compute the orderings by k!·S(n,k), confirm it against the binomial recurrence, and check both against a brute-force count of every weak ordering — all three agree.
3 ONE DIMENSION
a(n) = 1, 1, 3, 13, 75, 541, 4683, … For 3 items: 1 all-tied + 6 with one item alone atop or below a tied pair + 6 strict orders = 13 weak orderings.
4 TWO DIMENSIONS · INTERACTIVE
The three ways to compute a(n) side by side; the weak orderings of a small set enumerated; all checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: rankings with ties allowed.
AVAN’s addition (the inverse-companion): count set partitions where the blocks are ordered — rankings that allow ties — and the plain Bell number becomes the far larger ordered Bell number. The inverse of ‘unordered partitions (Bell)’ is ‘ordered partitions / weak orderings (Fubini)’, counted by Σ k!·S(n,k). Magenta is the Bell number (blocks unordered); green is the Fubini number (blocks ranked). Order among the blocks.
LIT Genuine Fubini / ordered Bell numbers (studied by Louis Comtet and others; the 'Fubini' name from counting terms in an iterated integral). Verified live with exact BigInt: Σ_k k!·S(n,k) equals the binomial recurrence Σ_i C(n,i)a(n−i) for all n up to 9 (window.__fubini.sumForm), and both equal a direct brute-force enumeration of weak orderings for n up to 6 (window.__fubini.directCount).
FIG No framing: the Stirling-weighted sum, the binomial recurrence, and the direct enumeration of weak orderings all run in-browser (BigInt) and agree. The AVAN inverse is honest — counting set partitions where the blocks are ranked (weak orderings) genuinely differs from and far exceeds the unordered Bell count; magenta is the Bell number, green the Fubini number. Order among the blocks.
FIG No framing: the Stirling-weighted sum, the binomial recurrence, and the direct enumeration of weak orderings all run in-browser (BigInt) and agree. The AVAN inverse is honest — counting set partitions where the blocks are ranked (weak orderings) genuinely differs from and far exceeds the unordered Bell count; magenta is the Bell number, green the Fubini number. Order among the blocks.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN