Grammar Specification of a Combinatorial Class

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
- テンソル解析
- パッケージ
- ファンクションアドバイザ
- ベキ級数
- ベクトル解析
- 不活性関数
- 代数
- 因数分解と方程式解法
- 基本数学
- 基本的な注意事項
- 変分法
- 変換
- 幾何
- 微分代数方程式
- 微分幾何学
- 微分方程式
- 微積分
- 数
- 数値計算
- 数学関数
- 数論
- 最適化
- 特殊関数
- 統計
- 線形代数
- 群論
- 評価
- 論理
- 過去のパッケージ
- 金融
- 離散数学
  - グラフ理論
  - 組合せ論
    - combinat
    - gfun
    - Permutations
    - 変換
    - 組み合わせ構造
    - !
    - binomial
    - MOLS
  - 総和と差分方程式
  - 離散変換
  - OrderBasis コマンド
  - 積
- piecewise examples
- グラフ電卓
- 区分関数
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
エラーメッセージガイド
マニュアル
数学アプリ

Description

•	A combinatorial class is either an elementary class, or is built from simpler classes with "constructors." The elementary classes are Epsilon, which represents an object of size zero, and Atom, which represents an object of size one. The available constructors are listed in the following table.


Epsilon	object of size 0
Atom	object of size 1 (Z is a predefined atom)
Union(A,B,...)	disjointunion of the classes A,B, ...
Prod(A,B,...)	partitional product of the classes A, B, ...
Set(A)	all sets with repetitions whose elements are in A
PowerSet(A)	all sets without repetitions whose elements are in A
Sequence(A)	all sequences of elements of A
Cycle(A)	all directed cycles of elements of A
Subst(A,B)	B-objects whose atoms are replaced by A-objects

•	For the constructors Set, PowerSet, Sequence, and Cycle, it is possible to add restrictions on the cardinality. For example, means all nonempty sets whose elements are in , means all sequences of at most three elements of , and means all directed cycles of five elements from .

None of these constructors accept an object that generates Epsilon as an argument. In some cases, no cardinality restriction or with any constructor but PowerSet, such a grammar generates an infinite number of objects of size 0, whereas this system is for classes with a finite number of members of each size.

In the others, PowerSet or or , while there are only a finite number of objects of size 0, the current system does not handle grammars with such Epsilon productions.

•	In , neither nor may produce objects of size 0.

•	A specification is a set of productions of the form , where is the name of the class being defined, and is an expression involving elementary classes, constructors, and other classes. For example, the following table lists specifications of some well-known combinatorial classes.

When the labeling type is 'labeled'.


	nonplane trees
	plane binary trees
	plane general trees
	permutations
	set partitions
	nonplane ternary trees
	hierarchies
	3-balanced hierarchies
	surjections
	functional graphs

When the labeling type is 'unlabeled'.


	integer partition
	binary sequences
	necklaces
	rooted unlabeled trees
	nonplane binary trees
	nonplane ternary trees
	unlabeled hierarchies
	random mappings patterns
	2-3 trees
	integer partitions without
	repetition

•	It is possible to use Epsilon as a way of marking certain objects without affecting their size.

•	There are also predefined structures, for example, combinations, built into the system. For more information, see combstruct[structures].

Examples

Generate words on two letters of the form .

(1)

If you do not need the derivation structure, remove it.

(2)

To model series and parallel circuits of resistors, a parallel circuit is made up of two or more resistors in series, and a series circuit is made up of two or more parallel circuits.

(3)

However, you cannot tell from this answer which resistors are in series and which are in parallel. In this case, you obtain more information about the derivation by using Epsilon tags.