HurwitzNumber

Hurwitz's number for the partition: (n + c - 2)! n^(c-3) times the product of p^p / (p-1)! over its c parts p, n being its size.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001763via IntegerPartition✓↗

HurwitzNumber(partition)Hurwitz's number for the partition: (n + c - 2)! n^(c-3) times the product of p^p / (p-1)! over its c parts p, n being its size.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition and 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

0 if Length(_x) < 1 else (1 if Sum(_x) + (Length(_x) - 2) == 0 else Product(Range(
                                                                                   1,
                                                                                   Sum(_x) +
                                                                                   (Length(_x) - 2),
                                                                                 ))) *
                         Sum(_x) ^ Max(Length(_x) - 3, 0) * Product(Map(
                                                                         Function(
                                                                                   _x[i] ^ _x[i],
                                                                                   i,
                                                                                 ),
                                                                         Range(1, Length(_x)),
                                                                       )) / (Sum(_x) ^ Max(3 - Length(_x), 0) * Product(Map(
                                                                             Function(
                                                                                       1 if _x[i] - 1 ==
                                                                                            0 else Product(Range(1, _x[i] - 1)),
                                                                                       i,
                                                                                     ),
                                                                             Range(1, Length(_x)),
                                                                           )))

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{HurwitzNumber}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$6\,881\,280$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{HurwitzNumber}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$7\,348\,320$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{HurwitzNumber})$Out$1$

Known value · FindStat St001763

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{HurwitzNumber})$Out$1$

Known value · FindStat St001763

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{HurwitzNumber})$Out$3$

Known value · FindStat St001763

About

Examples	5
Crosswalk	FindStat