HookNumber

The number of hooks in the decomposition of the tree into hooks: a node with the chains of left children and of right children below it, the subtrees hanging off those chains decomposed in turn.

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HookNumber(tree)The number of hooks in the decomposition of the tree into hooks: a node with the chains of left children and of right children below it, the subtrees hanging off those chains decomposed in turn.

Typed signatures

Type	Package	Applies
(binary_tree) -> number	combinatorics	always

Domain: Tree statistics

Details

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

Apply(Function(
                0 if t in Integers else 1 + self(self, t[1], 1) + self(
                                                                        self,
                                                                        t[2],
                                                                        2,
                                                                      ) if mode ==
                                                                          0 else self(self, t[2], 0) +
                                                                                self(self, t[1], 1) if mode == 1 else self(self, t[1], 0) +
                                                                                                   self(self, t[2], 2),
                self,
                t,
                mode,
              ), Function(0 if t in Integers else 1 + self(self, t[1], 1) +
                                                     self(self, t[2], 2) if mode == 0 else self(
                                                                              self,
                                                                              t[2],
                                                                              0,
                                                                            ) +
                                                                        self(
                                                                              self,
                                                                              t[1],
                                                                              1,
                                                                            ) if mode ==
                                                                            1 else self(self, t[1], 0) +
                                                                                  self(self, t[2], 2), self, t, mode), _x, 0)

Examples

Basic #

the binary tree $[[.,.],[[.,.],.]]$

live ↗

In$\mathrm{HookNumber}({\operatorname{BinaryTree}}(\bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, \bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, 0\bigr\rbrack\bigr\rbrack))$Out$2$

Properties #

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryTree}}(\bigl\lbrack0, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack), \text{HookNumber})$Out$1$

Known value · FindStat St000568

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryTree}}(\bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, 0\bigr\rbrack), \text{HookNumber})$Out$1$

Known value · FindStat St000568

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryTree}}(\bigl\lbrack0, \bigl\lbrack0, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack\bigr\rbrack), \text{HookNumber})$Out$1$

Known value · FindStat St000568

About

Examples	4
Crosswalk	FindStat