SmallestMissingPart

The smallest positive integer that is not a part.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000759via IntegerPartition✓↗St001449via PlanePartition✓↗

SmallestMissingPart(partition)The smallest positive integer that is not a part.

Typed signatures

Type	Package	Applies
(integer_partition | plane_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

1 if Length(_x) == 0 else Min(Filter(
                                      Range(1, Length(_x) + 1),
                                      Function(Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       _x[i] ==
                                                                       k,
                                                                       i,
                                                                     ),
                                                           )) == 0, k),
                                    ))

on PlanePartition, over _x

1 if Length(Flatten(_x)) < 1 else Apply((reading: list<integer>) => Apply(
                                                                           (lengths: list<integer>) => Min(Map(
                                                                                                                Function(k if Max(Sum(Map(Function(1 if e == k else 0, e), reading)), 0) ==
                                                                                                                              0 else Length(reading) + 1, k),
                                                                                                                Range(1, Length(reading) + 1, 1),
                                                                                                              )),
                                                                           Map(
                                                                                Function(Length(r), r),
                                                                                _x,
                                                                              ),
                                                                         ), Flatten(_x))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000759

FindStat's value for [2]

live ↗

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

Known value · FindStat St000759

FindStat's value for [3]

live ↗

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

Known value · FindStat St000759

About

Examples	5
Crosswalk	FindStat ×2