CartanDeterminant

The determinant of the matrix of the minimums of the distinct parts: the least distinct part times the gaps between consecutive distinct parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001571via IntegerPartition✓↗

CartanDeterminant(partition)The determinant of the matrix of the minimums of the distinct parts: the least distinct part times the gaps between consecutive distinct 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 Product(Map(Function(_x[i] -
                                              (0 if i == Length(_x) else _x[i +
                                                                            1]), i), Filter(
                                                            Range(1, Length(_x)),
                                                            Function(i ==
                                                                     Length(_x) || _x[i] >
                                                                         _x[i +
                                                                            1], i),
                                                          )))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001571

FindStat's value for [2]

live ↗

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

Known value · FindStat St001571

FindStat's value for [3]

live ↗

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

Known value · FindStat St001571

About

Examples	5
Crosswalk	FindStat