InversionsOfCyclicEmbedding

The inversions of the cycle (p(1), ..., p(n), n + 1), as a permutation of n + 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001074via Permutation✓↗

InversionsOfCyclicEmbedding(p)The inversions of the cycle (p(1), ..., p(n), n + 1), as a permutation of n + 1.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation 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 Permutation, over _x

0 if Length(_x) < 1 else Apply(
                                (q: list<integer>) => Sum(Map(
                                                               Function(Count(Filter(
                                                                                      Range(i + 1, Length(_x) + 1),
                                                                                      Function(q[i] > q[j], j),
                                                                                    )), i),
                                                               Range(
                                                                      1,
                                                                      Length(_x),
                                                                    ),
                                                             )),
                                Map(Function(
                                              _x[1] if v == Length(_x) + 1 else Sum(Map(Function(
                                                                                                  (Length(_x) + 1 if m == Length(_x) else _x[m + 1]) if _x[m] == v else 0,
                                                                                                  m,
                                                                                                ), Range(1, Length(_x)))),
                                              v,
                                            ), Range(1, Length(_x) + 1)),
                              )

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

In$\mathrm{InversionsOfCyclicEmbedding}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$6$

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001074

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001074

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St001074

About

Examples	5
Crosswalk	FindStat