THE FOLD / LOOT / THE VAULT / THE LAH NUMBERS
THE LAH NUMBERS
numbers linking the rising and falling factorials
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Lah numbers L(n,k) are the exact exchange rate between the two natural kinds of factorial. The rising factorial x(n) = x(x+1)…(x+n-1) and the falling factorial (x)k = x(x-1)…(x-k+1) each build a ‘staircase’ product, one climbing and one descending. The unsigned Lah numbers convert one into the other: x(n) = ∑k L(n,k) (x)k, with the clean closed form L(n,k) = C(n-1, k-1) · n!/k!. Combinatorially, L(n,k) counts the ways to split n labelled items into k non-empty ordered lists. They sit between the Stirling numbers as the ‘both-ordered’ case, and satisfy L(n,1) = n!, L(n,n) = 1.
LIT verified live: L(n,k) = C(n-1,k-1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x(n) = ∑k L(n,k)(x)k holds exactly for a range of x and n (window.__lah). FIG no framing; the Lah closed form and the factorial identity are computed independently in-browser.
LIT verified live: L(n,k) = C(n-1,k-1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x(n) = ∑k L(n,k)(x)k holds exactly for a range of x and n (window.__lah). FIG no framing; the Lah closed form and the factorial identity are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — the loot: a table of numbers that is the exact currency between rising and falling factorials. AVAN (AI) built the instrument: the Lah closed form, the L(n,1)/L(n,n) edges, and the rising-to-falling identity.
Credit as content: Ivo Lah (1954). The weave: David names the exchange rate; I confirm x(n) = ∑ L(n,k)(x)k.
Credit as content: Ivo Lah (1954). The weave: David names the exchange rate; I confirm x(n) = ∑ L(n,k)(x)k.
3 ONE DIMENSION
The Lah triangle: L(n,k) = C(n−1,k−1)·n!/k!, with n! down the left edge and 1 down the right.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n and x; the rising factorial x^(n) is rebuilt as Σ L(n,k)(x)_k from falling factorials.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the rising factorial x^(n), rebuilt from falling factorials.
AVAN’s addition (the inverse-companion): don’t recompute the climbing product — convert the descending one. The inverse of ‘the rising factorial x(n)’ is ‘∑k L(n,k) (x)k, Lah-weighted falling factorials’. Magenta are the Lah-weighted falling-factorial pieces; green is the rising factorial they sum to. The exchange rate between two staircases.
LIT Genuine Lah numbers (Ivo Lah, 1954). Verified live: L(n,k) = C(n−1,k−1)·n!/k! is an integer with L(n,1) = n! and L(n,n) = 1, and the identity x^(n) = Σ_k L(n,k)(x)_k holds exactly for a range of x and n (window.__lah.formOk, .idOk).
FIG No framing; the Lah closed form and the factorial identity are computed independently in-browser. The AVAN inverse is honest — instead of recomputing the climbing product, convert the descending one: the inverse of 'the rising factorial x^(n)' is 'Σ_k L(n,k)(x)_k, Lah-weighted falling factorials'. Magenta are the Lah-weighted falling-factorial pieces; green is the rising factorial they sum to. The exchange rate between two staircases.
FIG No framing; the Lah closed form and the factorial identity are computed independently in-browser. The AVAN inverse is honest — instead of recomputing the climbing product, convert the descending one: the inverse of 'the rising factorial x^(n)' is 'Σ_k L(n,k)(x)_k, Lah-weighted falling factorials'. Magenta are the Lah-weighted falling-factorial pieces; green is the rising factorial they sum to. The exchange rate between two staircases.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN