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THE MOTZKIN
paths that may rest — Catalan hiding under the flats
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Motzkin numbers 1, 1, 2, 4, 9, 21, 51, 127, … count the lattice paths from (0,0) to (n,0) using up, down, and level steps that never dip below the axis — the ‘lazy cousin’ of Dyck paths, which forbid the level step. They also count non-crossing chords on a circle and unary–binary trees.
The recurrence Mₙ = Mn−1 + Σ Mk·Mn−2−k (start with a level step, or an up…down arc enclosing a sub-path), and they tie to Catalan by Mₙ = Σk C(n,2k)·Catk.
LIT verified live: the recurrence matches a brute count of Motzkin paths for n=0…10, and the Catalan relation Mₙ = Σ C(n,2k)·Catk holds exactly (window.__motzkin). FIG no framing; exact path counting.
The recurrence Mₙ = Mn−1 + Σ Mk·Mn−2−k (start with a level step, or an up…down arc enclosing a sub-path), and they tie to Catalan by Mₙ = Σk C(n,2k)·Catk.
LIT verified live: the recurrence matches a brute count of Motzkin paths for n=0…10, and the Catalan relation Mₙ = Σ C(n,2k)·Catk holds exactly (window.__motzkin). FIG no framing; exact path counting.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hello-world — the first path allowed to pause as well as rise and fall. Motzkin paths are exactly that: Dyck paths that may rest. AVAN (AI) built the instrument: the recurrence, the brute path enumeration, the Catalan relation.
Credit as content: Theodore Motzkin (1948). The weave: David names the first resting path; I count Motzkin paths two ways and reveal the Catalan skeleton hiding under the level steps.
Credit as content: Theodore Motzkin (1948). The weave: David names the first resting path; I count Motzkin paths two ways and reveal the Catalan skeleton hiding under the level steps.
3 ONE DIMENSION
A Motzkin path: up, down, and flat steps from the axis back to the axis, never dipping below. The flat step is the only difference from a Dyck path — it lets the walk rest.
4 TWO DIMENSIONS · INTERACTIVE
Pick n. The instrument computes Mₙ by the recurrence and by brute-counting paths, confirms they agree, checks the Catalan relation, and draws a few Motzkin paths.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: Motzkin paths rising, falling, and resting back to the axis.
AVAN’s addition (the inverse-companion): adding the level step is a coarsening — every Motzkin path is a Dyck path with flats inserted, so Motzkin numbers interpolate between Catalan (all up/down) and the trivial all-flat path. The relation Mₙ = Σk C(n,2k)·Catk says it exactly: choose which 2k of the n steps form the non-flat Dyck skeleton, fill the rest with flats. The inverse of ‘a richer step set’ is ‘a Catalan path wearing flats.’ Magenta is the flat steps — the new freedom to rest; green is the Dyck skeleton underneath. Allowing the walk to pause reveals Catalan hiding inside every resting path.
LIT Genuine Motzkin numbers (Motzkin 1948). Verified live: M_n = M_{n-1} + sum_{k} M_k M_{n-2-k} matches a brute count of up/down/level paths staying >=0 for n=0..10, and M_n = sum_k C(n,2k) Cat_k holds exactly (window.__motzkin.recMatchesBrute && .catalanRelation); M = 1,1,2,4,9,21,51,127,323.
FIG No framing: the recurrence, the brute path enumeration, and the Catalan relation all compute in-browser and agree exactly. The AVAN inverse is honest — every Motzkin path is a Dyck path with flats inserted, and M_n = sum C(n,2k)Cat_k literally chooses which 2k steps form the non-flat Dyck skeleton; magenta is the level steps, green the Dyck skeleton.
FIG No framing: the recurrence, the brute path enumeration, and the Catalan relation all compute in-browser and agree exactly. The AVAN inverse is honest — every Motzkin path is a Dyck path with flats inserted, and M_n = sum C(n,2k)Cat_k literally chooses which 2k steps form the non-flat Dyck skeleton; magenta is the level steps, green the Dyck skeleton.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN