SmallestNonRepeatedInteger

The smallest positive integer that is not a part at least twice.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001568via IntegerPartition✓↗

SmallestNonRepeatedInteger(partition)The smallest positive integer that is not a part at least twice.

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.
• The integer need not be a part: it is the least k >= 1 with fewer than two parts equal to k.

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,
                                                                     ),
                                                           )) < 2, k),
                                    ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St001568

FindStat's value for [3]

live ↗

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

Known value · FindStat St001568

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{SmallestNonRepeatedInteger})$Out$1$

Known value · FindStat St001568

About

Examples	5
Crosswalk	FindStat