WeylCatalanNumber
The Catalan number of the type, ∏ (d + h) / d over the degrees d and the Coxeter number h.
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St000138via CartanType✓WeylCatalanNumber(cartanType)The Catalan number of the type, ∏ (d + h) / d over the degrees d and the Coxeter number h.Typed signatures
| Type | Package | Applies |
|---|---|---|
(cartan_type) -> number | combinatorics | always |
Details
- Defined over
CartanTypeas an expression in_x, evaluated by compute-engine — the definition IS the implementation. - Takes a
CartanTypeand 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.
Definition
Epsil, as the implementation runs it.
on CartanType, over _x
Apply(
(ds: list<integer>) => Product(Map(
Function(1 * Max(ds) + (d - 0), d),
ds,
)) / Product(ds) if (Product(Map(
Function(1 * Max(ds) +
(d - 0), d),
ds,
)) + 1) %
2 == (Product(Map(
Function(1 * Max(ds) +
(d - 0), d),
ds,
)) %
2 + 1) % 2 else 0 / 0,
Range(2, _x[2] + 1, 1) if _x[1] == "A" else Map(
Function(2i, i),
Range(1, _x[2]),
) if _x[1] == "B" else Map(
Function(2i, i),
Range(1, _x[2]),
) if _x[1] == "C" else Join(
Filter(
Map(Function(2i, i), Range(1, _x[2] - 1)),
Function(e < _x[2], e),
),
[_x[2]],
Filter(
Map(Function(2i, i), Range(1, _x[2] - 1)),
Function(e >= _x[2], e),
),
) if _x[1] == "D" else ([2, 5, 6, 8, 9, 12] if _x[2] == 6 else [2, 6, 8, 10, 12, 14, 18] if _x[2] == 7 else [2, 8, 12, 14, 18, 20, 24, 30]) if _x[1] == "E" else [2, 6, 8, 12] if _x[1] == "F" else [2, 6],
)Examples
Properties #
About
| Examples | 5 |
|---|---|
| Crosswalk | FindStat |