PairsOfDistinctParts

Pairs of parts that differ in size: the degree of the polynomial counting semistandard tableaux of the stretched shape.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000175via IntegerPartition✓↗

PairsOfDistinctParts(partition)Pairs of parts that differ: the degree of the polynomial that counts semistandard tableaux of the stretched shape.

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 Sum(Map(Function(Count(Filter(
                                                        Range(1, Length(_x)),
                                                        Function(j < i && _x[j] >
                                                                          _x[i], j),
                                                      )), i), Range(
                                                                    1,
                                                                    Length(_x),
                                                                  )))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000175

FindStat's value for [2]

live ↗

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

Known value · FindStat St000175

FindStat's value for [3]

live ↗

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

Known value · FindStat St000175

About

Examples	5
Crosswalk	FindStat