Hag

The excedance tops less their positions, plus the inversions among the excedance tops less those among the rest, plus the pairs k < i, over excedances i, with p(k) <= i.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001671via Permutation✓↗

Hag(p)The excedance tops less their positions, plus the inversions among the excedance tops less those among the rest, plus the pairs k < i, over excedances i, with p(k) <= i.

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.
• FindStat's Haglund hag. The inversions are within the subwords, as in Denert.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Max(Sum(Map(
                                      Function(_x[i] - i, i),
                                      Filter(
                                              Range(1, Length(_x)),
                                              Function(_x[i] > i, i),
                                            ),
                                    )), 0) + Sum(Map(
                                                     Function(Count(Filter(
                                                                            Range(i + 1, Length(_x)),
                                                                            Function(_x[j] > j &&
                                                                                     _x[i] > _x[j], j),
                                                                          )) if _x[i] >
                                                                               i else 0, i),
                                                     Range(1, Length(_x)),
                                                   )) + Sum(Map(
                                                                Function(Count(Filter(
                                                                                       Range(1, i - 1, 1),
                                                                                       Function(_x[k] <= i, k),
                                                                                     )), i),
                                                                Filter(
                                                                        Range(1, Length(_x)),
                                                                        Function(
                                                                                  _x[i] > i,
                                                                                  i,
                                                                                ),
                                                                      ),
                                                              )) -
                         Sum(Map(Function(
                                           Count(Filter(
                                                         Range(
                                                                i + 1,
                                                                Length(_x),
                                                              ),
                                                         Function(_x[j] <= j &&
                                                                  _x[i] > _x[j], j),
                                                       )) if _x[i] <= i else 0,
                                           i,
                                         ), Range(1, Length(_x))))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001671

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001671

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001671

About

Examples	5
Crosswalk	FindStat