HuangRhoadesMajorIndex

The major index of the flattened permutation, plus for each block ending before the next begins the elements so far beyond each block's first.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000747via SetPartition✓↗

HuangRhoadesMajorIndex(partition)The major index of the flattened permutation, plus for each block ending before the next begins the elements so far beyond each block's first.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition 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.
• Huang and Rhoades' variant (Ordered set partitions and the 0-Hecke algebra, eq. 2.7).

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

(0 if Length(Flatten(_x)) < 2 else Sum(Map(
                                            Function(i if Flatten(_x)[i] >
                                                          Flatten(_x)[i + 1] else 0, i),
                                            Range(1, Length(Flatten(_x)) - 1),
                                          ))) +
(0 if Length(_x) < 2 else Sum(Map(
                                   Function(
                                             Sum(Map(
                                                      Function(
                                                                Length(_x[t]) -
                                                                1 if t <= i else 0,
                                                                t,
                                                              ),
                                                      Range(1, Length(_x)),
                                                    )) if Last(_x[i]) <
                                                         _x[i + 1][1] else 0,
                                             i,
                                           ),
                                   Range(1, Length(_x) - 1),
                                 )))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

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

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

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

Properties #

FindStat's value for {{1,2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2]]), \text{HuangRhoadesMajorIndex})$Out$0$

Known value · FindStat St000747

FindStat's value for {{1,2,3}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{HuangRhoadesMajorIndex})$Out$0$

Known value · FindStat St000747

FindStat's value for {{1},{2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1], [2]]), \text{HuangRhoadesMajorIndex})$Out$0$

Known value · FindStat St000747

About

Examples	5
Crosswalk	FindStat