PermutationBounce

The sum of n - b over the bounce points b: the position of 1 in p, then the greatest position among the entries up to one past the last.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000133via Permutation✓↗

PermutationBounce(p)The sum of n - b over the bounce points b: the position of 1 in p, then the greatest position among the entries up to one past the last.

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(
                                (v: list<integer>) => Max(Sum(Map(
                                                                   Function(Length(_x) -
                                                                            Fold(
                                                                                  Function(Max(Map(
                                                                                                    Function(v[j], j),
                                                                                                    Range(1, w + 1, 1),
                                                                                                  )) if w < Length(_x) else Length(_x), w, s),
                                                                                  v[1],
                                                                                  Range(1, t, 1),
                                                                                ), t),
                                                                   Range(
                                                                          0,
                                                                          Length(_x) -
                                                                          1,
                                                                          1,
                                                                        ),
                                                                 )), 0),
                                Map(
                                     Function(
                                               Sum(Map(
                                                        Function(
                                                                  j if _x[j] ==
                                                                       m else 0,
                                                                  j,
                                                                ),
                                                        Range(1, Length(_x)),
                                                      )),
                                               m,
                                             ),
                                     Range(1, Length(_x)),
                                   ),
                              )

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000133

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000133

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000133

About

Examples	5
Crosswalk	FindStat