BruhatTranspositionCount

Transpositions that are at most p in Bruhat order.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001558via Permutation✓↗

BruhatTranspositionCount(p)Transpositions that are at most p in Bruhat order.

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.
• Equal to the inversions exactly when p avoids 4231 and 3412.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001558

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001558

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001558

About

Examples	5
Crosswalk	FindStat