Tier

Positions a with a larger entry, then the value p(a) - 1, to its right.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000779via Permutation✓↗

Tier(p)Positions a with a larger entry, then the value p(a) - 1, to its right.

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

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{Tier}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000779

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000779

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2, 3]), \text{Tier})$Out$0$

Known value · FindStat St000779

About

Examples	5
Crosswalk	FindStat