InvariantOrientedCycleCount

The cyclic permutations that a permutation of this cycle type fixes under conjugation: none unless the parts are all equal to some d, in which case totient(d) d^(m-1) (m-1)! for m parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000510via IntegerPartition✓↗

InvariantOrientedCycleCount(partition)The cyclic permutations that a permutation of this cycle type fixes under conjugation: none unless the parts are all equal to some d, in which case totient(d) d^(m-1) (m-1)! for m parts.

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 Count(Filter(Range(1, Max(_x)), Function(k * Max(_x) /
                                                                  LCM(
                                                                       k,
                                                                       Max(_x),
                                                                     ) == 1, k))) *
                         Max(_x) ^ (Length(_x) - 1) *
                         (1 if Length(_x) - 1 == 0 else Product(Range(
                                                                       1,
                                                                       Length(_x) -
                                                                       1,
                                                                     ))) if Max(_x) == _x[Length(_x)] else 0

Examples

Basic #

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

live ↗

In$\mathrm{InvariantOrientedCycleCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$0$

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

live ↗

In$\mathrm{InvariantOrientedCycleCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$0$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000510

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{InvariantOrientedCycleCount})$Out$2$

Known value · FindStat St000510

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{InvariantOrientedCycleCount})$Out$2$

Known value · FindStat St000510

About

Examples	5
Crosswalk	FindStat