EdgeOrbitCount

Orbits of the complete graph's edges under a permutation with these cycle lengths: the sum of half the parts rounded down, plus the sum of gcd(part, part) over pairs.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001982via IntegerPartition✓↗

EdgeOrbitCount(partition)Orbits of the complete graph's edges under a permutation with these cycle lengths: the sum of half the parts rounded down, plus the sum of gcd(part, part) over pairs.

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 Sum(Map(
                                  Function((_x[i] - _x[i] % 2) / 2, i),
                                  Range(1, Length(_x)),
                                )) + Sum(Map(Function(Sum(Map(
                                                              Function(
                                                                        _x[i] *
                                                                        _x[j] /
                                                                        LCM(
                                                                             _x[i],
                                                                             _x[j],
                                                                           ) if j <
                                                                            i else 0,
                                                                        j,
                                                                      ),
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                            )), i), Range(
                                                                          1,
                                                                          Length(_x),
                                                                        )))

Examples

Basic #

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

live ↗

In$\mathrm{EdgeOrbitCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$7$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{EdgeOrbitCount})$Out$0$

Known value · FindStat St001982

FindStat's value for [2]

live ↗

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

Known value · FindStat St001982

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{EdgeOrbitCount})$Out$1$

Known value · FindStat St001982

About

Examples	5
Crosswalk	FindStat