LargestMissingPart

The largest integer below the largest part that is not a part; 0 if every smaller integer is.

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LargestMissingPart(partition)The largest integer below the largest part that is not a part; 0 if every smaller integer is.

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.
• 0 counts as a candidate, so the value is 0 when every smaller positive integer is a part.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Max(Map(
                                  Function(k if Count(Filter(
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                              Function(_x[i] ==
                                                                       k, i),
                                                            )) == 0 else 0, k),
                                  Range(1, Max(_x)),
                                ))

Examples

Basic #

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

live ↗

In$\mathrm{LargestMissingPart}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$3$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001392

FindStat's value for [2]

live ↗

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

Known value · FindStat St001392

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{LargestMissingPart})$Out$2$

Known value · FindStat St001392

About

Examples	5
Crosswalk	FindStat