CatalanFactorCount
The factors in the Catalan factorization: the critical steps, and the middle Dyck word if it is not empty.
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St000876via BinaryWord✓CatalanFactorCount(word)The factors in the Catalan factorization: the critical steps, and the middle Dyck word if it is not empty.Typed signatures
| Type | Package | Applies |
|---|---|---|
(binary_word) -> number | combinatorics | always |
Details
- Defined over
BinaryWordas an expression in_x, evaluated by compute-engine — the definition IS the implementation. - Takes a
BinaryWordand nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer. - FindStat's St000876. The middle word is not empty when the height reaches its lowest value more than once.
Definition
Epsil, as the implementation runs it.
on BinaryWord, over _x
0 if Length(_x) < 1 else Apply(
(h: list<integer>) => Apply((low: integer) => h[Length(_x) + 1] -
2low +
(1 if Max(Sum(Map(
Function(1 if h[j + 1] ==
low else 0, j),
Range(0, Length(_x), 1),
)), 0) >
1 else 0), Min(h)),
Apply((a_init: list<integer>) => Fold(
(
a: list<integer>,
b,
) => Apply(
(a_next: list<integer>) => a_next,
Join(a, [a[Length(a)] +
(2 * _x[b] -
1)]),
),
a_init,
Range(1, Length(_x)),
), [0]),
)Examples
Properties #
About
| Examples | 4 |
|---|---|
| Crosswalk | FindStat |