NormalizedLeafLabelSum

The sum of the leaf labels of the increasing binary tree of p, less n.

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NormalizedLeafLabelSum(p)The sum of the leaf labels of the increasing binary tree of p, less n.

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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000726

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000726

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

live ↗

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

Known value · FindStat St000726

About

Examples	6
Crosswalk	FindStat