THE FOLD / RESPAWN / HARD RESET / THE INVOLUTION
THE INVOLUTION
self-inverse permutations counted by a recurrence
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Involutions are the permutations that are their own inverse: apply one twice and you’re back where you started (σ² = identity). Structurally they are made only of fixed points and 2-cycles — every element is either left alone or swapped with exactly one partner. The number of involutions of n elements is the telephone number T(n) (also the number of ways to pair up n telephones with some left unconnected): 1, 1, 2, 4, 10, 26, 76, 232, 764, …. It satisfies the recurrence T(n) = T(n-1) + (n-1)·T(n-2), and by the RSK correspondence it also counts the standard Young tableaux with n cells.
LIT verified live: a brute count of the permutations σ with σ² = identity equals the telephone number T(n) = T(n-1) + (n-1)T(n-2) and the explicit sum ∑k n!/(2kk!(n-2k)!) for every n from 0 to 8 (window.__involution). FIG no framing; the brute involution count, the recurrence, and the sum formula all run in-browser and agree.
LIT verified live: a brute count of the permutations σ with σ² = identity equals the telephone number T(n) = T(n-1) + (n-1)T(n-2) and the explicit sum ∑k n!/(2kk!(n-2k)!) for every n from 0 to 8 (window.__involution). FIG no framing; the brute involution count, the recurrence, and the sum formula all run in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hard-reset — the respawn: an involution applied twice is a hard reset to the identity, every swap undoing itself. AVAN (AI) built the instrument: the brute count of self-inverse permutations, the telephone recurrence, and the sum formula.
Credit as content: the telephone/involution numbers (Rothe, and via Young tableaux). The weave: David names the reset; I confirm the self-inverse permutations are counted by T(n).
Credit as content: the telephone/involution numbers (Rothe, and via Young tableaux). The weave: David names the reset; I confirm the self-inverse permutations are counted by T(n).
3 ONE DIMENSION
An involution of n elements — only fixed points (self-loops) and 2-cycles (swaps); applying it twice resets.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the brute count of σ²=id permutations is compared to T(n) and the sum formula.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the count of self-inverse permutations, T(n).
AVAN’s addition (the inverse-companion): don’t enumerate all σ with σ²=id — grow them. The inverse of ‘count the involutions of n’ is ‘T(n) = T(n-1) + (n-1)T(n-2)’: element n is either a fixed point or paired with one of the n-1 others. Magenta are the pairings and fixed points; green is the telephone number they total. Self-inverse permutations, counted by a recurrence.
LIT Genuine involution / telephone numbers (Rothe; via Young tableaux). Verified live: a brute count of permutations σ with σ²=identity equals the telephone number T(n)=T(n−1)+(n−1)T(n−2) and the sum Σ_k n!/(2^k k!(n−2k)!) for n=0..8 (window.__involution.ok, .sumOk).
FIG No framing; the brute involution count, the recurrence, and the sum formula all run in-browser and agree. The AVAN inverse is honest — instead of enumerating all σ with σ²=id, grow them: the inverse of 'count the involutions of n' is 'T(n)=T(n−1)+(n−1)T(n−2)': element n is either a fixed point or paired with one of the n−1 others. Magenta are the pairings and fixed points; green is the telephone number they total. Self-inverse permutations, counted by a recurrence.
FIG No framing; the brute involution count, the recurrence, and the sum formula all run in-browser and agree. The AVAN inverse is honest — instead of enumerating all σ with σ²=id, grow them: the inverse of 'count the involutions of n' is 'T(n)=T(n−1)+(n−1)T(n−2)': element n is either a fixed point or paired with one of the n−1 others. Magenta are the pairings and fixed points; green is the telephone number they total. Self-inverse permutations, counted by a recurrence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN