RefinementCount
The partitions that refine this one, whose parts subdivide to give theirs. Itself included.
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St000345via IntegerPartition✓RefinementCount(partition)The partitions that refine this one, whose parts subdivide to give theirs. Itself included.Typed signatures
| Type | Package | Applies |
|---|---|---|
(integer_partition) -> number | combinatorics | always |
Details
- Defined over
IntegerPartitionas an expression in_x, evaluated by compute-engine — the definition IS the implementation. - Takes a
IntegerPartitionand 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 IntegerPartition, over _x
1 if Length(_x) < 1 else 1 if Sum(_x) < 2 else Apply(
(mj: list<integer>) => Apply(
(ma: list<integer>) => Fold(
Function(Apply((fl: list<integer>) => Apply((ways: list<integer>) => Apply(
(ap: list<integer>) => Apply(
(cm: list<integer>) => Apply(
(sg: list<integer>) => Apply(
(kp: list<integer>) => [Map(Function(Apply(
(r: integer) => Apply((c: integer) => cm[(kp[r] - 1) *
Sum(_x) +
c], 1 +
(g - 1) %
Sum(_x)),
1 +
(g - 1 -
(g - 1) %
Sum(_x)) /
Sum(_x),
), g), Range(1, Length(kp) * Sum(_x))), [1], [a[3][1] + Length(kp)]],
Filter(
Range(1, Length(ap)),
Function(Count(Filter(
Range(1, Length(ap)),
Function(u < s &&
(Sum(Map(Function(Abs(cm[(s - 1) *
Sum(_x) +
k] -
cm[(u - 1) *
Sum(_x) +
k]), k), Range(1, Sum(_x)))) ==
0 if sg[s] == sg[u] else 1 == 0), u),
)) ==
0, s),
),
),
Map(Function(Sum(Map(Function(cm[(s - 1) *
Sum(_x) +
k] *
(k * k +
1), k), Range(1, Sum(_x)))), s), Range(1, Length(ap))),
),
Map(Function(Apply(
(s: integer) => Apply((c: integer) => Apply(
(e: integer) => Apply((q: integer) => fl[(e - 1) *
Sum(_x) +
c] -
(1 if c == mj[q] else 0) +
(1 if c == ma[q] else 0) +
(1 if c ==
mj[q] - ma[q] else 0), 1 +
(ap[s] - 1) %
Length(mj)),
1 +
(ap[s] - 1 -
(ap[s] - 1) %
Length(mj)) /
Length(mj),
), 1 +
(g - 1) %
Sum(_x)),
1 +
(g - 1 -
(g - 1) %
Sum(_x)) /
Sum(_x),
), g), Range(1, Length(ap) * Sum(_x))),
),
Filter(
Range(1, (Length(fl) -
Length(fl) % Sum(_x)) /
Sum(_x) *
Length(mj)),
Function(Apply(
(e: integer) => Apply((q: integer) => fl[(e - 1) *
Sum(_x) +
mj[q]] >=
1, 1 +
(t - 1) %
Length(mj)),
1 +
(t - 1 -
(t - 1) %
Length(mj)) /
Length(mj),
), t),
),
), a[2]), a[1]), a, b),
[Map(Function(Count(Filter(
Range(1, Length(_x)),
Function(_x[i] == k, i),
)), k), Range(1, Sum(_x))), [1], [1]],
Range(1, Sum(_x) - Length(_x), 1),
),
Flatten(Map(Function(Map(
Function(pa, pa),
Range(1, (pj -
pj % 2) /
2),
), pj), Range(2, Sum(_x))), 1),
),
Flatten(Map(
Function(Map(
Function(pj, pa),
Range(1, (pj -
pj % 2) /
2),
), pj),
Range(
2,
Sum(_x),
),
), 1),
)[3][1]Examples
Properties #
About
| Examples | 5 |
|---|---|
| Crosswalk | FindStat |