THE FOLD / CO-OP / THE HANDOFF / THE SCHRODER
THE SCHRODER
Catalan with a flat step — super-Catalan path counts
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The (large) Schröder numbers count the lattice paths from (0,0) to (2n,0) using up-steps (1,1), down-steps (1,−1), and flat steps (2,0), never dipping below the axis. They are the Catalan numbers with a flat step allowed — a ‘super-Catalan’ count — and satisfy a clean convolution recurrence. The sequence is 1, 2, 6, 22, 90, 394, 1806, …
LIT verified live: the recurrence equals a brute enumeration of all such Schröder paths for n ≤ 5, and the values match the known large-Schröder sequence 1,2,6,22,90,394 (window.__schroder). FIG no framing; exact combinatorics.
LIT verified live: the recurrence equals a brute enumeration of all such Schröder paths for n ≤ 5, and the values match the known large-Schröder sequence 1,2,6,22,90,394 (window.__schroder). FIG no framing; exact combinatorics.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-handoff — each count handed up from smaller ones by splitting a path at its first return to the axis, beside the Delannoy numbers. Schröder is that flat-step cousin. AVAN (AI) built the instrument: the convolution recurrence, the brute path enumeration, the known-sequence check.
Credit as content: Ernst Schröder (1870). The weave: David names the handoff; I split each path at its first return and multiply the sub-counts, and confirm the totals equal an exhaustive enumeration.
Credit as content: Ernst Schröder (1870). The weave: David names the handoff; I split each path at its first return and multiply the sub-counts, and confirm the totals equal an exhaustive enumeration.
3 ONE DIMENSION
A Schröder path: up, down, or a flat double-step, staying at or above the axis and ending on it. The flat step is what separates Schröder from the strictly up/down Catalan paths.
4 TWO DIMENSIONS · INTERACTIVE
The Schröder numbers by recurrence and by brute path count; a sample path is drawn for the chosen n.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the recurrence building each Schröder count from smaller ones.
AVAN’s addition (the inverse-companion): a path decomposes at its first return to the axis into a smaller path inside and a smaller path after — giving a convolution recurrence, so the count builds from products of smaller counts. The inverse of ‘enumerate the paths’ is ‘split at the first return and multiply the sub-counts.’ Magenta is the exponential path list; green is the convolution recurrence. Large Schröder 1, 2, 6, 22, 90, 394 — Catalan with a flat step. (Kin to the-delannoy, the-motzkin, the-catalan.)
LIT Genuine large Schröder numbers (Schröder 1870). Verified live: the recurrence S(n)=(3(2n-1)S(n-1)-(n-2)S(n-2))/(n+1) equals a brute enumeration of all up/down/flat Schröder paths (staying >=0) for n<=5, and the values are 1,2,6,22,90,394 (window.__schroder.recMatchesBrute && .matchesKnown).
FIG No framing: the convolution recurrence, the brute path enumeration, and the known-sequence check run in-browser and agree exactly. The AVAN inverse is honest — a path splits at its first return to the axis into an inside and an after path, giving a convolution recurrence, so counts build from products of smaller counts; magenta is the exponential path list, green the recurrence. Large Schröder is Catalan with a flat step. Kin to the-delannoy, the-motzkin, the-catalan.
FIG No framing: the convolution recurrence, the brute path enumeration, and the known-sequence check run in-browser and agree exactly. The AVAN inverse is honest — a path splits at its first return to the axis into an inside and an after path, giving a convolution recurrence, so counts build from products of smaller counts; magenta is the exponential path list, green the recurrence. Large Schröder is Catalan with a flat step. Kin to the-delannoy, the-motzkin, the-catalan.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN