SimsunDoubleDescents

Double descents in the restrictions of p to the values 1..k, over all k, counted with multiplicity.

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SimsunDoubleDescents(p)Double descents in the restrictions of p to the values 1..k, over all k, counted with multiplicity.

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) < 3 else Sum(Map(Function(Sum(Map(
                                                   Function(Sum(Map(
                                                                     Function(Max(
                                                                                   0,
                                                                                   Min(Join(
                                                                                             [Length(_x) + 1],
                                                                                             Map(
                                                                                                  Function(_x[m], m),
                                                                                                  Filter(
                                                                                                          Range(1, Length(_x)),
                                                                                                          Function(m > i &&
                                                                                                                   m < k &&
                                                                                                                   m != j, m),
                                                                                                        ),
                                                                                                ),
                                                                                           )) -
                                                                                   _x[i],
                                                                                 ) if _x[i] > _x[j] &&
                                                                                     _x[j] > _x[k] else 0, k),
                                                                     Filter(
                                                                             Range(1, Length(_x)),
                                                                             Function(k > j, k),
                                                                           ),
                                                                   )), j),
                                                   Filter(
                                                           Range(1, Length(_x)),
                                                           Function(j > i, j),
                                                         ),
                                                 )), i), Range(1, Length(_x))))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{SimsunDoubleDescents}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{SimsunDoubleDescents}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{SimsunDoubleDescents})$Out$0$

Known value · FindStat St000367

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{SimsunDoubleDescents})$Out$0$

Known value · FindStat St000367

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{SimsunDoubleDescents})$Out$0$

Known value · FindStat St000367

About

Examples	5
Crosswalk	FindStat