THE FOLD / LOOT / THE HOARD / THE HOOK LENGTH
THE HOOK LENGTH
count Young tableaux as n! over a product of hooks
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The hook length formula counts the standard Young tableaux of a shape λ — the ways to fill the cells of a Young diagram with 1…n so numbers increase along every row and down every column. Astonishingly, the count is just n! divided by the product of the hook lengths.
Each cell’s hook is itself, plus the cells to its right (the arm), plus the cells below it (the leg); multiply all these hooks and divide n! by the product. A global count of intricate fillings collapses to one clean product — and these counts are the dimensions of the irreducible representations of the symmetric group.
LIT verified live: n! / (product of hook lengths) equals a brute count of standard Young tableaux for every partition of n=1…7 (window.__hooklength). FIG no framing; exact.
Each cell’s hook is itself, plus the cells to its right (the arm), plus the cells below it (the leg); multiply all these hooks and divide n! by the product. A global count of intricate fillings collapses to one clean product — and these counts are the dimensions of the irreducible representations of the symmetric group.
LIT verified live: n! / (product of hook lengths) equals a brute count of standard Young tableaux for every partition of n=1…7 (window.__hooklength). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — counting the arrangements of a treasure without laying every one out. The hook length formula is that count, in one product. AVAN (AI) built the instrument: the hook-length computation, the factorial-over-product formula, the brute standard-tableau count.
Credit as content: J. S. Frame, Gilbert de B. Robinson & Robert M. Thrall (1954). The weave: David names the hoard; I compute each cell’s hook, form n! over their product, and confirm it equals a direct enumeration of the valid tableaux.
Credit as content: J. S. Frame, Gilbert de B. Robinson & Robert M. Thrall (1954). The weave: David names the hoard; I compute each cell’s hook, form n! over their product, and confirm it equals a direct enumeration of the valid tableaux.
3 ONE DIMENSION
A cell’s hook: the cell itself (1), plus its arm (cells to the right), plus its leg (cells below). The hook length is 1 + arm + leg — a purely local quantity per cell.
4 TWO DIMENSIONS · INTERACTIVE
A Young diagram with each cell’s hook length shown. The instrument forms n! / (product of hooks) and checks it against a brute count of standard Young tableaux of that shape.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the diagram’s hooks, whose product divides n! to count all tableaux.
AVAN’s addition (the inverse-companion): an intractable-looking count — all valid fillings — becomes a simple product because the tableaux carry deep symmetry. The formula is exact and reveals that the number of tableaux is n! divided by a purely local geometric quantity: one hook per cell. The inverse of ‘enumerate every valid tableau’ is ‘multiply one number per cell.’ And these counts fλ satisfy Σλ (fλ)² = n! — tying the hooks to the RSK bijection between permutations and tableau-pairs. Magenta is the exponentially-many tableaux never enumerated; green is the product of hooks that counts them. Global counting from local geometry.
LIT Genuine hook length formula (Frame, Robinson & Thrall 1954). Verified live: n! / (product of hook lengths) equals a brute-force count of standard Young tableaux for every partition of n=1..7 (window.__hooklength.formulaMatchesBrute); shape [3,2] gives 5 tableaux.
FIG No framing: the hook computation, the factorial-over-product formula, and the brute standard-tableau count run in-browser and agree exactly. The AVAN inverse is honest — a global count of tableaux becomes a product of one local hook per cell, and these counts f^lambda satisfy sum(f^lambda)^2 = n! (the RSK identity); magenta is the un-enumerated tableaux, green the hooks. Ties to the-rsk.
FIG No framing: the hook computation, the factorial-over-product formula, and the brute standard-tableau count run in-browser and agree exactly. The AVAN inverse is honest — a global count of tableaux becomes a product of one local hook per cell, and these counts f^lambda satisfy sum(f^lambda)^2 = n! (the RSK identity); magenta is the un-enumerated tableaux, green the hooks. Ties to the-rsk.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN