SimionSchmidtPreimageSize

The permutations sent to p by the Simion-Schmidt map onto the 321-avoiding permutations.

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SimionSchmidtPreimageSize(p)The permutations sent to p by the Simion-Schmidt map onto the 321-avoiding permutations.

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.
• That map keeps the left-to-right maxima and puts the other entries in increasing order, so p needs those increasing; each other position then chooses among the smaller values left.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

1 if Length(_x) < 1 else Product(Map(
                                      Function(Max(Map(
                                                        Function(_x[m], m),
                                                        Range(1, i),
                                                      )) - i + 1, i),
                                      Filter(
                                              Range(1, Length(_x)),
                                              Function(_x[i] < Max(Map(
                                                                        Function(_x[m], m),
                                                                        Range(1, i),
                                                                      )), i),
                                            ),
                                    )) if Sum(Map(
                                                  Function(Count(Filter(
                                                                         Range(1, Length(_x)),
                                                                         Function(j > i &&
                                                                                  _x[j] < _x[i], j),
                                                                       )), i),
                                                  Filter(
                                                          Range(1, Length(_x)),
                                                          Function(_x[i] <
                                                                   Max(Map(
                                                                            Function(_x[m], m),
                                                                            Range(1, i),
                                                                          )), i),
                                                        ),
                                                )) == 0 else 0

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000124

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000124

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000124

About

Examples	5
Crosswalk	FindStat