THE FOLD / CO-OP / THE HANDOFF / THE DELANNOY
THE DELANNOY
king-path counting — Pascal with a third, diagonal term
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Delannoy numbers D(m,n) count the lattice paths from (0,0) to (m,n) using three step types: east (1,0), north (0,1), and the diagonal (1,1) — a king’s moves. That extra diagonal step is the whole story: D(m,n) = D(m−1,n) + D(m,n−1) + D(m−1,n−1), a Pascal-like recurrence with a third term. The central values D(n,n) are 1, 3, 13, 63, 321, 1683, …
LIT verified live: the recurrence equals a brute enumeration of all king-paths for m,n ≤ 5, and the central Delannoy numbers match the known sequence (window.__delannoy). FIG no framing; exact combinatorics.
LIT verified live: the recurrence equals a brute enumeration of all king-paths for m,n ≤ 5, and the central Delannoy numbers match the known sequence (window.__delannoy). FIG no framing; exact combinatorics.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-handoff — each cell’s count is handed up from its three predecessors, corner to corner across the grid. The Delannoy recurrence is that chain of handoffs. AVAN (AI) built the instrument: the three-term recurrence table, the brute king-path enumeration, the central-sequence check.
Credit as content: Henri Delannoy (1895). The weave: David names the handoff; I build each count from its west, south, and diagonal neighbours and confirm it equals an exhaustive path count.
Credit as content: Henri Delannoy (1895). The weave: David names the handoff; I build each count from its west, south, and diagonal neighbours and confirm it equals an exhaustive path count.
3 ONE DIMENSION
A path reaches (m,n) from one of three neighbours: west (an east step), south (a north step), or the diagonal (a diagonal step). So its count is the sum of those three predecessors’ counts.
4 TWO DIMENSIONS · INTERACTIVE
The Delannoy grid; each cell shows D(m,n), the diagonal on the main axis giving the central Delannoy numbers, all checked against brute path counts.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: each cell’s count summed from its three predecessors.
AVAN’s addition (the inverse-companion): a path to (m,n) arrives from one of three neighbours — west, south, or the diagonal — so D(m,n) = D(m−1,n) + D(m,n−1) + D(m−1,n−1); the count builds from three smaller counts, no enumeration. The inverse of ‘list every path’ is ‘each cell’s count is the sum of its three predecessors.’ Magenta is the exponentially-many paths never listed; green is the triangular recurrence. The diagonal step is the third term that distinguishes Delannoy from Pascal.
LIT Genuine Delannoy numbers (Delannoy 1895). Verified live: the three-term recurrence D(m,n)=D(m-1,n)+D(m,n-1)+D(m-1,n-1) equals a brute enumeration of all (east/north/diagonal) king-paths for m,n<=5, and the central Delannoy numbers are 1,3,13,63,321,1683 (window.__delannoy.recurrenceMatchesBrute && .centralMatchesKnown).
FIG No framing: the three-term recurrence table, the brute king-path enumeration, and the central-sequence check run in-browser and agree exactly. The AVAN inverse is honest — a path to (m,n) arrives from its west, south, or diagonal neighbor, so each count is the sum of three predecessors, no enumeration; magenta is the exponentially-many paths never listed, green the recurrence. The diagonal step is the third term distinguishing Delannoy from Pascal.
FIG No framing: the three-term recurrence table, the brute king-path enumeration, and the central-sequence check run in-browser and agree exactly. The AVAN inverse is honest — a path to (m,n) arrives from its west, south, or diagonal neighbor, so each count is the sum of three predecessors, no enumeration; magenta is the exponentially-many paths never listed, green the recurrence. The diagonal step is the third term distinguishing Delannoy from Pascal.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN