THE FOLD / LOOT / THE BOUNTY / THE Q-BINOMIAL
THE Q-BINOMIAL
counting subspaces with a q-analog of the binomial
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Gaussian binomial coefficient [n, k]q is the q-analog of the ordinary binomial: replace each integer m by its q-version 1 + q + … + qm−1. It has a product form and, remarkably, two combinatorial meanings. As a polynomial in q it is Σ q(inversions) over binary words with k ones; and when q is a prime power, [n, k]q counts exactly the number of k-dimensional subspaces of the vector space Fqn. Setting q = 1 recovers the ordinary C(n, k). One formula, three faces: product, inversion-sum, subspace count.
LIT verified live: the product form equals the inversion-sum for q in {2, 3, 5} and n up to 7 (exact BigInt), and equals the brute-force count of k-subspaces of F2n for n up to 4 (window.__qbinomial). FIG no framing; three computations of one quantity, cross-checked.
LIT verified live: the product form equals the inversion-sum for q in {2, 3, 5} and n up to 7 (exact BigInt), and equals the brute-force count of k-subspaces of F2n for n up to 4 (window.__qbinomial). FIG no framing; three computations of one quantity, cross-checked.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — a single count that pays out three ways: a product, a sum over inversions, and the number of subspaces you can carve from Fqn. That triple payout is the bounty. AVAN (AI) built the instrument: the q-product, the inversion-sum over binary words, and the exhaustive subspace count over a finite field.
Credit as content: Carl Friedrich Gauss (the coefficients); the subspace interpretation is classical finite-field combinatorics. The weave: David names the-bounty; I compute [n,k]q as a product, as a sum of q raised to inversion counts, and as the number of k-subspaces of Fqn — and confirm all three agree.
Credit as content: Carl Friedrich Gauss (the coefficients); the subspace interpretation is classical finite-field combinatorics. The weave: David names the-bounty; I compute [n,k]q as a product, as a sum of q raised to inversion counts, and as the number of k-subspaces of Fqn — and confirm all three agree.
3 ONE DIMENSION
[3,1]2 = (2³−1)/(2−1) = 7 = number of lines through 0 in F2³. [4,2]2 = 35. As a q-polynomial, [n,k]q = Σ qinversions; at q=1 it is C(n,k).
4 TWO DIMENSIONS · INTERACTIVE
The three faces of [n,k]q — product, inversion-sum, subspace count; all cross-checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a binomial that counts subspaces.
AVAN’s addition (the inverse-companion): take the ordinary “choose k of n” and deform it by q — and the deformed count becomes the number of k-dimensional subspaces of a finite vector space. The inverse of ‘count k-element subsets (q=1)’ is ‘count k-dimensional subspaces (q = prime power).’ Magenta is the plain binomial C(n,k); green is the q-binomial [n,k]q. Choosing, deformed into subspaces.
LIT Genuine Gaussian binomial coefficient (Carl Friedrich Gauss); the subspace interpretation is classical finite-field combinatorics. Verified live with exact BigInt: the q-product Π(q^(n−i)−1)/(q^(k−i)−1) equals the inversion-sum Σ q^(inv) over binary words for q∈{2,3,5} and all n≤7 (window.__qbinomial.productEqualsInversion), and equals the exhaustive count of k-dimensional subspaces of F_2^n for n≤4 (window.__qbinomial.subspaceCount).
FIG No framing: the q-product, the inversion-sum over binary words, and the exhaustive subspace count over F_2^n all run in-browser and agree. The AVAN inverse is honest — deforming the ordinary 'choose k of n' by q turns it into the count of k-dimensional subspaces of a finite vector space (q=1 recovers subsets); magenta is the plain binomial C(n,k), green the q-binomial [n,k]_q. Choosing, deformed into subspaces.
FIG No framing: the q-product, the inversion-sum over binary words, and the exhaustive subspace count over F_2^n all run in-browser and agree. The AVAN inverse is honest — deforming the ordinary 'choose k of n' by q turns it into the count of k-dimensional subspaces of a finite vector space (q=1 recovers subsets); magenta is the plain binomial C(n,k), green the q-binomial [n,k]_q. Choosing, deformed into subspaces.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN