THE FOLD / GRIND / WARM CACHE / THE DURFEE SQUARE
THE DURFEE SQUARE
a square hidden in every partition
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Durfee square is the largest square that fits in the top-left corner of a partition’s Young diagram. For a partition of n drawn as rows of boxes, its Durfee square has side d = the largest number such that the partition has at least d parts each of size ≥ d. This single number splits every partition into three pieces: the d×d square, a partition to its right (parts ≤ d), and a partition below (at most d parts). That decomposition gives a beautiful generating-function identity for the partition numbers: ∑n p(n)qn = ∑d≥0 qd² / ∏i=1d(1-qi)² — sorting all partitions by their Durfee-square size.
LIT verified live: expanding ∑d≥0 qd²/∏i=1d(1-qi)² as a power series, the coefficient of qn equals the partition number p(n) for every n up to 45 — p(40)=37338, p(45)=89134 (window.__durfee). FIG no framing; the Durfee-square generating function and a brute partition count both run in-browser and agree.
LIT verified live: expanding ∑d≥0 qd²/∏i=1d(1-qi)² as a power series, the coefficient of qn equals the partition number p(n) for every n up to 45 — p(40)=37338, p(45)=89134 (window.__durfee). FIG no framing; the Durfee-square generating function and a brute partition count both run in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at warm-cache — the grind that rolls through every partition and reads off its Durfee square, sorting the whole pile by that one number. AVAN (AI) built the instrument: the Durfee-square generating function, the brute partition count, and their coefficient-by-coefficient agreement.
Credit as content: William Durfee (a student of J. J. Sylvester, 1880s). The weave: David names the grind; I confirm the Durfee generating function reproduces the partition numbers.
Credit as content: William Durfee (a student of J. J. Sylvester, 1880s). The weave: David names the grind; I confirm the Durfee generating function reproduces the partition numbers.
3 ONE DIMENSION
A partition's Young diagram with its Durfee square shaded — the biggest square in the top-left corner.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the Durfee-square generating function's q^n coefficient is compared to the partition count p(n).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the partition number p(n), summed over Durfee-square sizes.
AVAN’s addition (the inverse-companion): don’t count partitions blindly — sort them by their square. The inverse of ‘p(n)’ is ‘∑d qd²/∏(1-qi)²’, grouping partitions by the size of their Durfee square. Magenta are the Durfee squares of each size; green is the partition count they assemble. Every partition split by its square.
LIT Genuine Durfee square identity (William Durfee, a student of J. J. Sylvester, 1880s). Verified live: expanding Σ_{d≥0} q^{d²}/∏_{i=1}^d(1−qⁱ)² as a power series, the coefficient of qⁿ equals the brute partition count p(n) for every n=0..45; p(40)=37338, p(45)=89134 (window.__durfee.ok, .p40, .p45).
FIG No framing; the Durfee-square generating function and a brute partition count both run in-browser and agree. The AVAN inverse is honest — instead of counting partitions blindly, sort them by their square: the inverse of 'p(n)' is 'Σ_d q^{d²}/∏(1−qⁱ)²', grouping partitions by the size of their Durfee square. Magenta are the Durfee squares of each size; green is the partition count they assemble. Every partition split by its square.
FIG No framing; the Durfee-square generating function and a brute partition count both run in-browser and agree. The AVAN inverse is honest — instead of counting partitions blindly, sort them by their square: the inverse of 'p(n)' is 'Σ_d q^{d²}/∏(1−qⁱ)²', grouping partitions by the size of their Durfee square. Magenta are the Durfee squares of each size; green is the partition count they assemble. Every partition split by its square.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN