SmallestLeafLabel

The smallest label of a leaf of the increasing binary tree of p: the least peak, an end counting as a peak.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000725via Permutation✓↗

SmallestLeafLabel(p)The smallest label of a leaf of the increasing binary tree of p: the least peak, an end counting as a peak.

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 Min(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),
                                        ),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000725

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000725

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000725

About

Examples	5
Crosswalk	FindStat