AutomorphismGroupSize

The number of ways to permute identical children, at every vertex at once: the product, over the vertices, of the factorials of how many times each child tree occurs.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000417via OrderedTree✓↗

AutomorphismGroupSize(tree)The number of ways to permute identical children, at every vertex at once: the product, over the vertices, of the factorials of how many times each child tree occurs.

Typed signatures

Type	Package	Applies
(ordered_tree) -> number	combinatorics	always

Domain: Tree statistics

Details

• Defined over OrderedTree as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a OrderedTree 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 OrderedTree, over _x

Fold(
      Function(a * b, a, b),
      1,
      Map(
           Function(1 if Length(s) == 0 else Fold(
                                                   Function(a * b, a, b),
                                                   1,
                                                   Map(Function(1 if Count(Filter(
                                                                                   s,
                                                                                   Function(Length(Unique([k, u])) ==
                                                                                            1, k),
                                                                                 )) <
                                                                     2 else Product(Range(
                                                                                          1,
                                                                                          Count(Filter(s, Function(Length(Unique([k, u])) ==
                                                                                                                   1, k))),
                                                                                        )), u), Unique(s)),
                                                 ), s),
           Apply(
                  Function([t] if Length(t) == 0 else Join(
                                                            [t],
                                                            Fold(
                                                                  Function(
                                                                            Join(a, b),
                                                                            a,
                                                                            b,
                                                                          ),
                                                                  [],
                                                                  Map(Function(
                                                                                self(self, c),
                                                                                c,
                                                                              ), t),
                                                                ),
                                                          ), self, t),
                  Function([t] if Length(t) == 0 else Join(
                                                            [t],
                                                            Fold(
                                                                  Function(
                                                                            Join(a, b),
                                                                            a,
                                                                            b,
                                                                          ),
                                                                  [],
                                                                  Map(Function(
                                                                                self(self, c),
                                                                                c,
                                                                              ), t),
                                                                ),
                                                          ), self, t),
                  _x,
                ),
         ),
    )

Examples

Basic #

the ordered tree $[[],[[]],[]]$

live ↗

In$\mathrm{AutomorphismGroupSize}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack))$Out$2$

Properties #

FindStat's value for [[]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack), \text{AutomorphismGroupSize})$Out$1$

Known value · FindStat St000417

FindStat's value for [[[]]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack), \text{AutomorphismGroupSize})$Out$1$

Known value · FindStat St000417

FindStat's value for [[],[]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{OrderedTree}}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack), \text{AutomorphismGroupSize})$Out$2$

Known value · FindStat St000417

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Examples	4
Crosswalk	FindStat