PartsEqualToNextPart

Parts whose next smaller part has the same size: the number of parts less the distinct parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001091via IntegerPartition✓↗

PartsEqualToNextPart(partition)Parts whose next smaller part has the same size: the number of parts less the distinct parts.

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 Length(_x) - Count(Filter(
                                                    Range(1, Max(_x)),
                                                    Function(Count(Filter(
                                                                           Range(1, Length(_x)),
                                                                           Function(_x[i] == k, i),
                                                                         )) > 0, k),
                                                  ))

Examples

Basic #

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

live ↗

In$\mathrm{PartsEqualToNextPart}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$0$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001091

FindStat's value for [2]

live ↗

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

Known value · FindStat St001091

FindStat's value for [3]

live ↗

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

Known value · FindStat St001091

About

Examples	5
Crosswalk	FindStat