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# 数学代写|组合学代写Combinatorics代考|MATH4410 Admissible constructions and specifications

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The main goal of this section is to introduce formally the basic constructions that constitute the core of a specification language for combinatorial structures. This core is based on disjoint unions, also known as combinatorial sums, and on Cartesian products that we have just discussed. We shall augment it by the constructions of sequence, cycle, multiset, and powerset. A class is constructible or specifiable if it can be defined from primal elements by means of these constructions. The generating function of any such class satisfies functional equations that can be transcribed systematically from a specification; see Theorems I.1 and I.2, as well as Figure 14 at the end of this chapter for a summary.

I. 2.1. Basic constructions. First, we assume given a class $\mathcal{E}$ called the neutral class that consists of a single object of size 0 ; any such an object of size 0 is called a neutral object. and is usually denoted by symbols like $\epsilon$ or 1 . The reason for this terminology becomes clear if one considers the combinatorial isomorphism
$$\mathcal{A} \cong \mathcal{E} \times \mathcal{A} \cong \mathcal{A} \times \mathcal{E}$$

## 数学代写|组合学代写Combinatorics代考|Combinatorial sum (disjoint union)

Combinatorial sum (disjoint union). First consider combinatorial sum also known as disjoint union. The intent is to capture the union of disjoint sets, but without the constraint of any extraneous condition of disjointness. We formalize the (combinatorial) sum of two classes $\mathcal{B}$ and $\mathcal{C}$ as the union (in the standard set-theoretic sense) of two disjoint copies, say $\mathcal{B}^{\square}$ and $\mathcal{C}^{\diamond}$, of $\mathcal{B}$ and $\mathcal{C}$. A picturesque way to view the construction is as follows: first choose two distinct colours and repaint the elements of $\mathcal{B}$ with the $\square$-colour and the elements of $\mathcal{C}$ with the $\diamond$-colour. This is made precise by introducing two distinct “markers” $\square$ and $\diamond$, each a neutral object (i.e., of size zero); the disjoint union $\mathcal{B}+\mathcal{C}$ of $\mathcal{B}, \mathcal{C}$ is then defined as the standard set-theoretic union,
$$\mathcal{B}+\mathcal{C}:=({\square} \times \mathcal{B}) \cup({\diamond} \times \mathcal{C})$$
The size of an object in a disjoint union $\mathcal{A}=\mathcal{B}+\mathcal{C}$ is by definition inherited from its size in its class of origin, like in Equation (13). One good reason behind the definition adopted here is that the combinatorial sum of two classes is always welldefined. Furthermore, disjoint union is equivalent to a standard union whenever it is applied to disjoint sets.

## 组合学代写

$$\mathcal{A} \cong \mathcal{E} \times \mathcal{A} \cong \mathcal{A} \times \mathcal{E}$$

## 数学代写|组合学代写Combinatorics代考|Combinatorial sum(disjoint union)

$$\mathcal{B}+\mathcal{C}:=(\square \times \mathcal{B}) \cup(\diamond \times \mathcal{C})$$

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