THE FOLD / GRIND / THE CRON JOB / THE NARAYANA
THE NARAYANA
Catalan sliced by peaks — a refinement that sums back
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Narayana numbers N(n,k) refine the Catalan numbers: they count the Dyck paths of semilength n (balanced-parenthesis strings) that have exactly k peaks — a peak being an up-step immediately followed by a down-step, ‘()’. Summing over all k recovers the Catalan number: Σk N(n,k) = Cₙ.
The closed form is N(n,k) = (1/n)·C(n,k)·C(n,k−1), and the triangle 1; 1,1; 1,3,1; 1,6,6,1; 1,10,20,10,1 is symmetric (N(n,k)=N(n,n+1−k)) — Catalan sliced by a natural statistic.
LIT verified live: N(n,k) equals a brute count of Dyck paths with k peaks, and Σk N(n,k) equals the Catalan number, for n=1…8 (window.__narayana). FIG no framing; exact.
The closed form is N(n,k) = (1/n)·C(n,k)·C(n,k−1), and the triangle 1; 1,1; 1,3,1; 1,6,6,1; 1,10,20,10,1 is symmetric (N(n,k)=N(n,n+1−k)) — Catalan sliced by a natural statistic.
LIT verified live: N(n,k) equals a brute count of Dyck paths with k peaks, and Σk N(n,k) equals the Catalan number, for n=1…8 (window.__narayana). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — counting the peaks in a periodic climb, the ups-and-downs of a repeating run. The Narayana numbers are that peak-count, refining the Catalan total. AVAN (AI) built the instrument: the closed-form N(n,k), the brute peak-count of Dyck paths, the Catalan row-sum.
Credit as content: Tadepalli Venkata Narayana (1955). The weave: David names the cron-job; I count Dyck paths by their peaks, match them to the closed form, and show they sum back to Catalan.
Credit as content: Tadepalli Venkata Narayana (1955). The weave: David names the cron-job; I count Dyck paths by their peaks, match them to the closed form, and show they sum back to Catalan.
3 ONE DIMENSION
A Dyck path with its peaks marked — each place an up-step is immediately followed by a down-step. Two paths of the same length can have different peak counts; N(n,k) tallies how many have exactly k.
4 TWO DIMENSIONS · INTERACTIVE
Pick n. The Narayana row is computed by the closed form and by brute-counting Dyck paths by peaks; they agree, and the row sums to the Catalan number.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the symmetric Narayana triangle, refining each Catalan number.
AVAN’s addition (the inverse-companion): the Catalan number decomposes by a hidden statistic. Summing Narayana over k un-refines back to Catalan, so Narayana is Catalan ‘sliced by peaks,’ and the slicing is symmetric — peaks and valleys are interchangeable, giving N(n,k)=N(n,n+1−k). The inverse of ‘one Catalan count’ is ‘its refinement by a natural feature,’ and refinements like this expose the internal structure a single number hides. Magenta is the lumped Catalan total; green is the Narayana slices summing to it. One number, its histogram by peaks — the same Catalan objects, sorted by shape.
LIT Genuine Narayana numbers (Narayana 1955). Verified live: N(n,k)=(1/n)C(n,k)C(n,k-1) equals a brute count of Dyck paths of semilength n with exactly k peaks, and sum_k N(n,k) equals the Catalan number C_n, for n=1..8 (window.__narayana.matchesPeaks && .sumsToCatalan); N(4,k)=1,6,6,1.
FIG No framing: the closed form, the brute peak-count, and the Catalan row-sum run in-browser and agree exactly. The AVAN inverse is honest — Narayana is Catalan refined by the peak statistic (summing over k un-refines to Catalan), and the refinement is symmetric (peaks vs valleys, N(n,k)=N(n,n+1-k)); magenta is the lumped Catalan, green the Narayana slices. Ties to the-catalan and the-motzkin.
FIG No framing: the closed form, the brute peak-count, and the Catalan row-sum run in-browser and agree exactly. The AVAN inverse is honest — Narayana is Catalan refined by the peak statistic (summing over k un-refines to Catalan), and the refinement is symmetric (peaks vs valleys, N(n,k)=N(n,n+1-k)); magenta is the lumped Catalan, green the Narayana slices. Ties to the-catalan and the-motzkin.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN