DistinctOddPartsBelowLargestEvenPart

Distinct odd part sizes smaller than the largest even part.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001588via IntegerPartition✓↗

DistinctOddPartsBelowLargestEvenPart(partition)Distinct odd part sizes smaller than the largest even part.

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 Apply(
                                (e: number) => Count(Filter(
                                                             Range(1, Max(_x)),
                                                             Function(Count(Filter(
                                                                                    Range(1, Length(_x)),
                                                                                    Function(_x[i] == k, i),
                                                                                  )) >
                                                                      0 && k % 2 ==
                                                                          1 && k <
                                                                              e, k),
                                                           )),
                                Max(Map(Function(
                                                  _x[i] if _x[i] % 2 == 0 else 0,
                                                  i,
                                                ), Range(1, Length(_x)))),
                              )

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001588

FindStat's value for [2]

live ↗

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

Known value · FindStat St001588

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{DistinctOddPartsBelowLargestEvenPart})$Out$0$

Known value · FindStat St001588

About

Examples	5
Crosswalk	FindStat