NecklaceCount

The colorings of a cycle with these color multiplicities, up to rotation: the size's divisors d of the greatest common divisor of the parts, each contributing totient(d) multinomials, over the size.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001605via IntegerPartition✓↗

NecklaceCount(partition)The colorings of a cycle with these color multiplicities, up to rotation: the size's divisors d of the greatest common divisor of the parts, each contributing totient(d) multinomials, over the 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.
• FindStat's St001605 (the cyclic group); its title says colourings of a cycle.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Apply(
                                (g: number) => Sum(Map(
                                                        Function(
                                                                  Count(Filter(
                                                                                Range(1, d),
                                                                                Function(k * d /
                                                                                         LCM(k, d) ==
                                                                                         1, k),
                                                                              )) *
                                                                  (1 if Sum(_x) /
                                                                        d == 0 else Product(Range(1, Sum(_x) / d))) /
                                                                  Product(Map(
                                                                               Function(
                                                                                         1 if _x[i] / d ==
                                                                                              0 else Product(Range(1, _x[i] / d)),
                                                                                         i,
                                                                                       ),
                                                                               Range(1, Length(_x)),
                                                                             )) if g % d == 0 else 0,
                                                                  d,
                                                                ),
                                                        Range(1, g),
                                                      )),
                                Fold(
                                      Function(a * b / LCM(a, b), a, b),
                                      _x[1],
                                      Rest(_x),
                                    ),
                              ) / Sum(_x)

Examples

Basic #

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

live ↗

In$\mathrm{NecklaceCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$15$

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

live ↗

In$\mathrm{NecklaceCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$20$

Properties #

FindStat's value for [3]

live ↗

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

Known value · FindStat St001605

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{NecklaceCount})$Out$1$

Known value · FindStat St001605

FindStat's value for [5]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([5]), \text{NecklaceCount})$Out$1$

Known value · FindStat St001605

About

Examples	5
Crosswalk	FindStat