RajchgotIndex

The sum over positions i of n - i less the longest increasing subsequence of the larger entries after p(i).

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RajchgotIndex(p)The sum over positions i of n - i less the longest increasing subsequence of the larger entries after p(i).

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• The degree of the Grothendieck polynomial, after Pechenik, Speyer and Weigandt.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Max(Sum(Map(Function(Length(_x) - i - Apply(
                                                                      (v: list<integer>) => Count(Fold(Function(Apply(
                                                                                                                       Function(Join(a, [b]) if k > Count(a) else ReplaceAt(a, k, b), k),
                                                                                                                       Count(Filter(a, Function(t < b, t))) +
                                                                                                                       1,
                                                                                                                     ), a, b), [], v)),
                                                                      Map(
                                                                           Function(_x[j], j),
                                                                           Filter(
                                                                                   Range(1, Length(_x)),
                                                                                   Function(j > i &&
                                                                                            _x[j] > _x[i], j),
                                                                                 ),
                                                                         ),
                                                                    ), i), Range(1, Length(_x)))), 0)

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{RajchgotIndex}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$2$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001759

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001759

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001759

About

Examples	6
Crosswalk	FindStat