PartsExceedingOnes

Parts larger than the number of parts equal to 1.

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PartsExceedingOnes(partition)Parts larger than the number of parts equal to 1.

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] > Count(Filter(
                                                                      Range(
                                                                             1,
                                                                             Length(_x),
                                                                           ),
                                                                      Function(_x[i] ==
                                                                               1, i),
                                                                    )), i),
                                     ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000473

FindStat's value for [2]

live ↗

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

Known value · FindStat St000473

FindStat's value for [3]

live ↗

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

Known value · FindStat St000473

About

Examples	5
Crosswalk	FindStat