SymplecticDimensionRankThree

The dimension of the irreducible representation of Sp(6) with this highest weight; 0 if the partition has more than 3 parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000716via IntegerPartition✓↗

SymplecticDimensionRankThree(partition)The dimension of the irreducible representation of Sp(6) with this highest weight; 0 if the partition has more than 3 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

1 if Length(_x) < 1 else 0 if Length(_x) > 3 else ((_x[1] if Length(_x) >= 1 else 0) +
                                                   3) *
                                                  ((_x[2] if Length(_x) >= 2 else 0) +
                                                   2) *
                                                  ((_x[3] if Length(_x) >= 3 else 0) +
                                                   1) * (((_x[1] if Length(_x) >=
                                                                   1 else 0) + 3) ^
                                                        2 - ((_x[2] if Length(_x) >=
                                                                      2 else 0) +
                                                            2) ^ 2) *
                                                  (((_x[1] if Length(_x) >= 1 else 0) +
                                                    3) ^ 2 - ((_x[3] if Length(_x) >=
                                                                       3 else 0) +
                                                             1) ^ 2) *
                                                  (((_x[2] if Length(_x) >= 2 else 0) +
                                                    2) ^ 2 - ((_x[3] if Length(_x) >=
                                                                       3 else 0) +
                                                             1) ^ 2) / 720

Examples

Basic #

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

live ↗

In$\mathrm{SymplecticDimensionRankThree}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1\,386$

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

live ↗

In$\mathrm{SymplecticDimensionRankThree}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$616$

Properties #

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{SymplecticDimensionRankThree})$Out$21$

Known value · FindStat St000716

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{SymplecticDimensionRankThree})$Out$56$

Known value · FindStat St000716

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{SymplecticDimensionRankThree})$Out$126$

Known value · FindStat St000716

About

Examples	5
Crosswalk	FindStat