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THE LAH
counting partitions into ordered lists
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Lah numbers L(n, k) count the ways to sort n labelled items into k non-empty ordered lists — where, unlike set partitions, the order within each list matters. They have a clean closed form, L(n, k) = C(n−1, k−1) · n! / k!, and a recurrence L(n, k) = L(n−1, k−1) + (n + k − 1) L(n−1, k). They are the exact coefficients that convert between the two great factorial bases: the rising factorial equals a Lah-weighted sum of falling factorials, x(n̅) = Σk L(n, k) x(k̲).
LIT verified live (exact BigInt): the closed form equals the recurrence for all n up to 12, and the basis-change identity x(n rising) = Σk L(n,k) x(k falling) holds for integer x (window.__lah). FIG no framing; exact big-integer combinatorics and polynomial identities.
LIT verified live (exact BigInt): the closed form equals the recurrence for all n up to 12, and the basis-change identity x(n rising) = Σk L(n,k) x(k falling) holds for integer x (window.__lah). FIG no framing; exact big-integer combinatorics and polynomial identities.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-inventory — not just which items go in which pile, but the order they are stacked in each pile; Lah numbers count exactly those ordered arrangements. That ordered inventory is the drop. AVAN (AI) built the instrument: the closed form C(n−1,k−1)·n!/k!, the two-term recurrence, and the rising/falling factorial basis-change identity.
Credit as content: Ivo Lah (1954). The weave: David names the-inventory; I compute L(n,k) by its closed form and by its recurrence and confirm they agree exactly, then verify the Lah numbers convert rising factorials into falling factorials — the connective tissue between the two factorial bases.
Credit as content: Ivo Lah (1954). The weave: David names the-inventory; I compute L(n,k) by its closed form and by its recurrence and confirm they agree exactly, then verify the Lah numbers convert rising factorials into falling factorials — the connective tissue between the two factorial bases.
3 ONE DIMENSION
L(n,k) = C(n−1,k−1)·n!/k!. L(3,1)=6 (one list, all orders of 3), L(3,2)=6, L(3,3)=1. Row sums are the “Lah” totals. They send x(n̅) to Σ L(n,k) x(k̲).
4 TWO DIMENSIONS · INTERACTIVE
The Lah triangle by closed form and recurrence; the rising/falling identity at a chosen x; both checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: partitions into ordered lists.
AVAN’s addition (the inverse-companion): count set partitions where the order inside each block matters, and the same numbers become the change-of-basis between rising and falling factorials. The inverse of ‘unordered blocks (Stirling)’ is ‘ordered lists (Lah)’, and they translate one factorial basis into the other. Magenta is the unordered Stirling count; green is the ordered Lah count. Order inside the blocks.
LIT Genuine Lah numbers (Ivo Lah, 1954). Verified live with exact BigInt: the closed form C(n−1,k−1)·n!/k! equals the two-term recurrence L(n,k)=L(n−1,k−1)+(n+k−1)L(n−1,k) for all n up to 12 (window.__lah.closedForm), and the basis-change identity x^(n rising) = Σ_k L(n,k)·x^(k falling) holds for integer x (window.__lah.factorialIdentity).
FIG No framing: the closed form, the recurrence, and the rising/falling factorial identity all run in-browser in exact big integers and agree. The AVAN inverse is honest — counting set partitions where the order inside each block matters (Lah) genuinely differs from the unordered Stirling count, and the same numbers translate one factorial basis into the other; magenta is the unordered Stirling count, green the ordered Lah count. Order inside the blocks.
FIG No framing: the closed form, the recurrence, and the rising/falling factorial identity all run in-browser in exact big integers and agree. The AVAN inverse is honest — counting set partitions where the order inside each block matters (Lah) genuinely differs from the unordered Stirling count, and the same numbers translate one factorial basis into the other; magenta is the unordered Stirling count, green the ordered Lah count. Order inside the blocks.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN