LengthCrank

The number of parts when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000146via IntegerPartition✓↗

LengthCrank(partition)The number of parts when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).

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.
• Crank has the largest part where this has the number of parts; FindStat's St000146 follows the code, which takes the length.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Length(_x) if Count(Filter(
                                                     Range(1, Length(_x)),
                                                     Function(_x[i] == 1, i),
                                                   )) == 0 else Count(Filter(
                                                                             Range(1, Length(_x)),
                                                                             Function(_x[i] >
                                                                                      Count(Filter(
                                                                                                    Range(1, Length(_x)),
                                                                                                    Function(_x[i] == 1, i),
                                                                                                  )), i),
                                                                           )) -
                                                               Count(Filter(
                                                                             Range(1, Length(_x)),
                                                                             Function(_x[i] == 1, i),
                                                                           ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000146

FindStat's value for [2]

live ↗

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

Known value · FindStat St000146

FindStat's value for [3]

live ↗

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

Known value · FindStat St000146

About

Examples	5
Crosswalk	FindStat