LongestIncreasingSubsequence

The length of a longest increasing subsequence.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLongest increasing subsequence↗WikidataQ4183855↗FindStatSt000062via Permutation✓↗Rosetta CodeLongest increasing subsequence↗

LongestIncreasingSubsequence(p)The length of a longest increasing subsequence.

CombinatoricaLength[LongestIncreasingSubsequence[p]]SageMathenumeratio_stat("Permutation", p, "LongestIncreasingSubsequence")

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.
• By patience sorting: the number of piles. Checked against the piles algorithm over every permutation of 1..6.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else 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), [], _x))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000062

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000062

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000062

About

Examples	6
Checked against	Combinatorica 3 agreeSageMath 6 agree
Crosswalk	WikipediaWikidataFindStatCombinatoricaSageMathRosetta Code

Implementation

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica returns a longest subsequence; ours is its length.

mappedsagepackages/oracle/src/mappings.ts