InsertionBumpsOfOne

One more than the longest decreasing subsequence of the entries before 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001390via Permutation✓↗

InsertionBumpsOfOne(p)One more than the longest decreasing subsequence of the entries before 1.

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.
• Inserting 1 bumps along the first column of the tableau built from the entries before it.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Apply(
                                (v: list<integer>) => 1 +
                                                      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), [], v)),
                                Map(
                                     Function(_x[i], i),
                                     Filter(
                                             Range(1, Length(_x)),
                                             Function(i < Sum(Map(
                                                                   Function(
                                                                             j if _x[j] == 1 else 0,
                                                                             j,
                                                                           ),
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                        ),
                                                                 )), i),
                                           ),
                                   ),
                              )

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001390

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001390

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001390

About

Examples	5
Crosswalk	FindStat