THE FOLD / GRIND / THE EPOCH / THE STIRLING CYCLES
THE STIRLING CYCLES
count permutations by cycles — the inverse of set partitions
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The unsigned Stirling numbers of the first kind c(n,k) count the permutations of n items with exactly k cycles — the mirror of the second kind, which counts set partitions into k blocks. The recurrence c(n,k) = (n−1)·c(n−1,k) + c(n−1,k−1): a new item joins an existing cycle in n−1 ways, or forms its own new cycle.
Rows sum to n! (every permutation has some number of cycles), and they are the coefficients of the rising factorial: x(x+1)(x+2)…(x+n−1) = Σk c(n,k)·xᵏ — the exact dual of the second kind’s falling-factorial identity.
LIT verified live: the recurrence matches a brute count of permutations by cycle number for n=1…7, each row sums to n!, and the rising-factorial identity holds exactly (window.__stirlingcycles). FIG no framing; exact counting.
Rows sum to n! (every permutation has some number of cycles), and they are the coefficients of the rising factorial: x(x+1)(x+2)…(x+n−1) = Σk c(n,k)·xᵏ — the exact dual of the second kind’s falling-factorial identity.
LIT verified live: the recurrence matches a brute count of permutations by cycle number for n=1…7, each row sums to n!, and the rising-factorial identity holds exactly (window.__stirlingcycles). FIG no framing; exact counting.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — beside its companion, the Stirling numbers of the second kind. Where the second kind groups a set into blocks, the first kind cycles a permutation; two ways to shatter n! into pieces. AVAN (AI) built the instrument: the recurrence, the brute cycle count, the rising-factorial identity.
Credit as content: James Stirling (Methodus Differentialis, 1730). The weave: David names the epoch; I count permutations by their cycles two ways and show the numbers are exactly the rising-factorial coefficients — the inverse table to the second kind.
Credit as content: James Stirling (Methodus Differentialis, 1730). The weave: David names the epoch; I count permutations by their cycles two ways and show the numbers are exactly the rising-factorial coefficients — the inverse table to the second kind.
3 ONE DIMENSION
A permutation drawn as its cycle diagram: follow each element to where it maps, and the arrows close into loops. The number of loops is the cycle count that c(n,k) tallies.
4 TWO DIMENSIONS · INTERACTIVE
Pick n. The instrument computes the first-kind row by recurrence and by brute-counting permutations by cycle number, confirms they agree, and checks the row sums to n! and the rising-factorial identity.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the first-kind triangle, each entry a count of permutations with k cycles.
AVAN’s addition (the inverse-companion): the two kinds of Stirling numbers are literally inverse matrices. The second kind converts ordinary powers to falling factorials; the (signed) first kind converts rising/falling factorials back to powers — and stacked as triangular matrices, they multiply to the identity. So ‘group a set into blocks’ and ‘cycle a permutation’ are not just parallel counts; as changes of basis between the power and factorial bases, each undoes the other. The inverse of the second kind’s table is the first kind’s. Magenta is the second-kind (set partitions); green is the first-kind (cycles) — two triangles that annihilate to I. The most natural ways to break n! apart are mutual inverses.
LIT Genuine unsigned Stirling numbers of the first kind (Stirling 1730). Verified live: c(n,k)=(n-1)c(n-1,k)+c(n-1,k-1) matches a brute count of permutations by cycle number for n=1..7, rows sum to n!, and x(x+1)...(x+n-1)=sum_k c(n,k) x^k holds for integer x (window.__stirlingcycles); c(5,k)=24,50,35,10,1.
FIG No framing: the recurrence, the brute cycle enumeration, the n! row sum, and the rising-factorial identity all compute in-browser and agree exactly. The AVAN inverse is honest — the first and second kind, as triangular change-of-basis matrices between the power and factorial bases, are genuine inverses (their product is the identity); magenta is second-kind (partitions), green first-kind (cycles).
FIG No framing: the recurrence, the brute cycle enumeration, the n! row sum, and the rising-factorial identity all compute in-browser and agree exactly. The AVAN inverse is honest — the first and second kind, as triangular change-of-basis matrices between the power and factorial bases, are genuine inverses (their product is the identity); magenta is second-kind (partitions), green first-kind (cycles).
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN