BoundaryLengthBeyondTrunk

The edges on the leftmost path or the rightmost path from the root, less the trunk they share.

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BoundaryLengthBeyondTrunk(tree)The edges on the leftmost path or the rightmost path from the root, less the trunk they share.

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

Apply(
       Function(0 if Length(t) == 0 else 1 + self(self, t[1]), self, t),
       Function(0 if Length(t) == 0 else 1 + self(self, t[1]), self, t),
       _x,
     ) + Apply(
               Function(0 if Length(t) == 0 else 1 + self(self, t[-1]), self, t),
               Function(0 if Length(t) == 0 else 1 + self(self, t[-1]), self, t),
               _x,
             ) - Apply(
                       Function(
                                 1 + self(self, t[1]) if Length(t) == 1 else 0,
                                 self,
                                 t,
                               ),
                       Function(
                                 1 + self(self, t[1]) if Length(t) == 1 else 0,
                                 self,
                                 t,
                               ),
                       _x,
                     )

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [[]]

live ↗

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

Known value · FindStat St000975

FindStat's value for [[[]]]

live ↗

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

Known value · FindStat St000975

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

live ↗

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

Known value · FindStat St000975

About

Examples	4
Crosswalk	FindStat