Runs

Maximal increasing runs — one more than the number of descents.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPermutation↗Permutation↗MathWorldPermutationRun↗Permutation↗WikidataQ161519↗Encyclopedia of MathematicsPermutation↗nLabpermutation↗Britannicatopic/permutation↗FindStatSt000325via Permutation✓↗St000470via Permutation✓↗

Runs(p)Maximal increasing runs — one more than the number of descents.

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

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 (0 if Length(_x) < 2 else Count(Filter(
                                                                 Range(
                                                                        1,
                                                                        Length(_x) -
                                                                        1,
                                                                      ),
                                                                 Function(_x[i] >
                                                                          _x[i +
                                                                             1], i),
                                                               ))) + 1

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000325

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000325

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000325

About

Examples	6
Checked against	Combinatorica 3 agreeSageMath 6 agree
Crosswalk	Wikipedia ×2MathWorld ×2WikidataEncyclopedia of MathematicsnLabBritannicaFindStat ×2CombinatoricaSageMath

Implementation

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica's Runs returns the runs themselves; ours is their count.

mappedsagepackages/oracle/src/mappings.ts