InvariantThreeElementSubsetCount

Invariant 3-element subsets: three fixed points, a fixed point and a 2-cycle, or a 3-cycle.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000512via IntegerPartition✓↗

InvariantThreeElementSubsetCount(partition)Invariant 3-element subsets: three fixed points, a fixed point and a 2-cycle, or a 3-cycle.

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, Length(_x)),
                                       Function(_x[i] == 1, i),
                                     )) * (Count(Filter(
                                                        Range(1, Length(_x)),
                                                        Function(_x[i] == 1, i),
                                                      )) - 1) * (Count(Filter(
                                                                              Range(1, Length(_x)),
                                                                              Function(_x[i] == 1, i),
                                                                            )) -
                                                                2) / 6 + Count(Filter(
                                                                                      Range(1, Length(_x)),
                                                                                      Function(_x[i] == 1, i),
                                                                                    )) *
                                                                        Count(Filter(
                                                                                      Range(1, Length(_x)),
                                                                                      Function(_x[i] == 2, i),
                                                                                    )) +
                         Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] == 3, i),
                                     ))

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{InvariantThreeElementSubsetCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$2$

Properties #

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{InvariantThreeElementSubsetCount})$Out$0$

Known value · FindStat St000512

FindStat's value for [3]

live ↗

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

Known value · FindStat St000512

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{InvariantThreeElementSubsetCount})$Out$0$

Known value · FindStat St000512

About

Examples	5
Crosswalk	FindStat