DepthIndex

The sum of n - i over the arcs i, less the depth of each element, plus the depth of each arc.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001094via SetPartition✓↗

DepthIndex(partition)The sum of n - i over the arcs i, less the depth of each element, plus the depth of each arc.

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.
• An element's depth is the arcs strictly over it; an arc's, the arcs strictly over it.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(Flatten(Map(Function(Most(b), b), _x))) == 0 else Length(Flatten(Map(
                                                                                  Function(Most(b), b),
                                                                                  _x,
                                                                                ))) *
                                                              Length(Flatten(_x)) - Length(Flatten(Map(
                                                                                     Function(Most(b), b),
                                                                                     _x,
                                                                                   ))) *
                                                                 (Length(Flatten(Map(
                                                                                      Function(Most(b), b),
                                                                                      _x,
                                                                                    ))) +
                                                                  1) / 2 + Sum(Map(
                                                                                 Function(Count(Filter(
                                                                                                        Range(1, Length(Flatten(Map(Function(Most(b), b), _x))), 1),
                                                                                                        Function(Flatten(Map(Function(Most(b), b), _x))[m] <
                                                                                                                 Flatten(Map(Function(Most(b), b), _x))[k] &&
                                                                                                                 Flatten(Map(Function(Rest(b), b), _x))[k] <
                                                                                                                 Flatten(Map(Function(Rest(b), b), _x))[m], m),
                                                                                                      )), k),
                                                                                 Range(
                                                                                        1,
                                                                                        Length(Flatten(Map(Function(Most(b), b), _x))),
                                                                                        1,
                                                                                      ),
                                                                               )) -
                                                              (Sum(Map(Function(
                                                                                 Last(b) - b[1],
                                                                                 b,
                                                                               ), _x)) -
                                                               Length(Flatten(Map(
                                                                                   Function(Most(b), b),
                                                                                   _x,
                                                                                 ))))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001094

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

live ↗

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

Known value · FindStat St001094

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

live ↗

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

Known value · FindStat St001094

About

Examples	5
Crosswalk	FindStat