THE FOLD / BOSS / THE GAUNTLET / THE EULERIAN NUMBERS
THE EULERIAN NUMBERS
count permutations by their climbs
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Eulerian numbers A(n,k) count the permutations of {1,…,n} with exactly k ascents (positions where the next element is larger). They form a triangle like Pascal’s but with a twist: A(n,k) = (k+1)·A(n−1,k) + (n−k)·A(n−1,k−1). Each row sums to n! (every permutation has some number of ascents), and the triangle is symmetric: A(n,k) = A(n,n−1−k), since reversing a permutation swaps ascents and descents.
LIT verified live: the recurrence matches a brute-force count of permutations by ascents for n ≤ 8, each row sums to n!, and the symmetry holds (window.__eulerian). FIG no framing; exact integer counts, checked against exhaustive enumeration.
LIT verified live: the recurrence matches a brute-force count of permutations by ascents for n ≤ 8, each row sums to n!, and the symmetry holds (window.__eulerian). FIG no framing; exact integer counts, checked against exhaustive enumeration.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gauntlet — run the whole permutation and count its climbs; the Eulerian triangle tallies how many runs have exactly k ascents. AVAN (AI) built the instrument: the Eulerian recurrence, the brute ascent-counter over all permutations, and the row-sum-equals-n! and symmetry checks.
Credit as content: Leonhard Euler (1755). The weave: David names the-gauntlet; I build the Eulerian triangle by its recurrence and confirm it equals the exhaustive count of permutations grouped by ascents — with each row summing to n! and mirror-symmetric, ascents and descents in balance.
Credit as content: Leonhard Euler (1755). The weave: David names the-gauntlet; I build the Eulerian triangle by its recurrence and confirm it equals the exhaustive count of permutations grouped by ascents — with each row summing to n! and mirror-symmetric, ascents and descents in balance.
3 ONE DIMENSION
A(n,k) = (k+1)A(n−1,k) + (n−k)A(n−1,k−1). Row 4: 1, 11, 11, 1 — permutations of 4 items with 0,1,2,3 ascents. They sum to 4! = 24.
4 TWO DIMENSIONS · INTERACTIVE
The Eulerian triangle; a chosen row checked against brute ascent-counting, with its row sum and symmetry.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: permutations tallied by their climbs.
AVAN’s addition (the inverse-companion): count permutations by their number of ascents not by enumerating all n! but by a Pascal-like recurrence — insert n into a smaller permutation, and it either extends an ascent block or splits one, giving A(n,k) = (k+1)A(n−1,k)+(n−k)A(n−1,k−1). The inverse of ‘list every permutation and tally ascents’ is ‘grow the count row by row.’ Magenta is the n! enumeration; green is the recurrence. Structure counted, not listed.
LIT Genuine Eulerian numbers (Leonhard Euler 1755). Verified live: the recurrence A(n,k)=(k+1)A(n−1,k)+(n−k)A(n−1,k−1) matches an exhaustive count of permutations of {1..n} grouped by ascents for n≤8 (window.__eulerian.recurrenceMatchesBrute), each row sums to n! (rowSumFactorial), and A(n,k)=A(n,n−1−k) (symmetry) — exact integer counts.
FIG No framing: the Eulerian recurrence, the brute ascent-counter over all permutations, and the row-sum-equals-n! and symmetry checks run in-browser with exact integers and agree. The AVAN inverse is honest — counting permutations by ascents via a Pascal-like recurrence (inserting n extends or splits an ascent block) genuinely replaces enumerating all n! permutations; magenta is the enumeration, green the recurrence. Structure counted, not listed.
FIG No framing: the Eulerian recurrence, the brute ascent-counter over all permutations, and the row-sum-equals-n! and symmetry checks run in-browser with exact integers and agree. The AVAN inverse is honest — counting permutations by ascents via a Pascal-like recurrence (inserting n extends or splits an ascent block) genuinely replaces enumerating all n! permutations; magenta is the enumeration, green the recurrence. Structure counted, not listed.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN