LargestSearchTreeLeafLabel

The largest label of a leaf of the binary search tree of p, inserting p(1), ..., p(n) in order.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000727via Permutation✓↗

LargestSearchTreeLeafLabel(p)The largest label of a leaf of the binary search tree of p, inserting p(1), ..., p(n) in order.

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(Map(
                                  Function(_x[i], i),
                                  Filter(
                                          Range(1, Length(_x)),
                                          Function(Apply(
                                                          (lo: integer) => Apply(
                                                                                  (hi: integer) => Count(Filter(
                                                                                                                 Range(1, Length(_x)),
                                                                                                                 Function(k > i &&
                                                                                                                          _x[k] > lo &&
                                                                                                                          _x[k] < hi, k),
                                                                                                               )) ==
                                                                                                   0,
                                                                                  Min(Join(
                                                                                            [Length(_x) + 1],
                                                                                            Map(
                                                                                                 Function(_x[j], j),
                                                                                                 Filter(
                                                                                                         Range(1, i),
                                                                                                         Function(_x[j] > _x[i], j),
                                                                                                       ),
                                                                                               ),
                                                                                          )),
                                                                                ),
                                                          Max(Join(
                                                                    [0],
                                                                    Map(
                                                                         Function(_x[j], j),
                                                                         Filter(
                                                                                 Range(1, i),
                                                                                 Function(_x[j] < _x[i], j),
                                                                               ),
                                                                       ),
                                                                  )),
                                                        ), i),
                                        ),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000727

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000727

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

live ↗

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

Known value · FindStat St000727

About

Examples	6
Crosswalk	FindStat