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# 数学代写|组合学代写Combinatorics代考|Labelled versus unlabelled enumeration

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## 数学代写|组合学代写Combinatorics代考|Labelled versus unlabelled enumeration

II. 2.2 Labelled versus unlabelled enumeration. Any labelled class $\mathcal{A}$ has an unlabelled counterpart $\widehat{\mathcal{A}}$ : objects in $\widehat{\mathcal{A}}$ are obtained from objects of $\mathcal{A}$ by ignoring the labels. This idea is formalized by identifying two labelled objects if there is an arbitrary relabelling (not just an order-consistent one, as has been used so far) that transforms one into the other. For an object of size $n$, each equivalence class contains a priori between 1 and $n$ ! elements. Thus:
PrOPOSITION II.1. The counts of a labelled class $\mathcal{A}$ and its unlabelled counterpart $\widehat{\mathcal{A}}$ are related by
$$\widehat{A}_n \leq A_n \leq n ! \widehat{A}_n \quad \text { or equivalently } \quad 1 \leq \frac{A_n}{\widehat{A}_n} \leq n !$$
EXAMPLE 5. Labelled and Unlabelled graphs. This phenomenon has been already encountered in our discussion of graphs (Figure 1). Let generally $G_n$ and $\widehat{G}_n$ be the number of graphs of size $n$ in the labelled and unlabelled case respectively. One finds for $n=1 \ldots 15$

The sequence $\left{\widehat{G}_n\right}$ constitutes EIS A000088, which can be obtained by an extension of methods of Chapter I; see [206, Ch. 4]. The sequence $\left{G_n\right}$ is determined directly by the fact that a graph of $n$ vertices can have each of the $\left(\begin{array}{l}n \ 2\end{array}\right)$ possible edges either present or not, so that
$$G_n=2^{\left(\begin{array}{c} n \ 2 \end{array}\right)}=2^{n(n-1) / 2} .$$

## 数学代写|组合学代写Combinatorics代考|Surjections and set partitions

II. 3.1. Surjections and set partitions. We examine classes
$$\mathcal{R}=\operatorname{SEQ}\left{\operatorname{SET}{\geq 1}{Z}\right} \quad \text { and } \quad \mathcal{S}=\operatorname{SET}\left{\operatorname{SET}{\geq 1}{Z}\right}$$
corresponding to sequences-of-sets $(\mathcal{R})$ and sets-of-sets $(\mathcal{S})$, or equivalently, sequences of urns and sets of urns, respectively. Such abstract specifications model very classical objects of discrete mathematics, namely surjections $(\mathcal{R})$ and set partitions $(\mathcal{S})$

Surjections with $r$ images. In elementary mathematics, a surjection from a set $A$ to a set $B$ is a function from $A$ to $B$ that assumes each value at least once (an onto mapping). Fix some integer $r \geq 1$ and let $\mathcal{R}_n^{(r)}$ denote the class of all surjections from the set $[1 \ldots n]$ onto $[1 \ldots r]$ whose elements are also called $r$-surjections.. Here is a particular object $\phi \in \mathcal{R}_9^{(5)}$ :

Note that, if $\phi(9)$ were 3 , then $\phi$ would not be a surjection. We set $\mathcal{R}^{(r)}=\bigcup_n \mathcal{R}n^{(r)}$ and proceed to compute the corresponding EGF, $R^{(r)}(z)$. First, let us observe that an $r$-surjection $\phi \in \mathcal{R}_n^{(r)}$ is determined by the ordered $r$-tuple formed with the collection of all preimage sets, $\left(\phi^{-1}(1), \phi^{-1}(2), \ldots, \phi^{-1}(r)\right)$, themselves disjoint nonempty sets of integers that cover the interval $[1 \ldots n]$. In the case of the surjection $\phi$ of (9), this alternative representation is $$\phi: \quad({2},{1,3},{4,6,8},{9},{5,7}) .$$ One has the combinatorial specification and EGF relation: (10) $\mathcal{R}^{(r)}=\mathrm{SEQ}_r{\mathcal{V}}, \mathcal{V}=\mathrm{SET}{\geq 1}{\mathcal{Z}} \quad \Longrightarrow \quad R^{(r)}(z)=\left(e^z-1\right)^r$.
There $\mathcal{V} \equiv \mathcal{U} \backslash{\epsilon}$ designates the class of urns $(\mathcal{U})$ that are nonempty, with EGF $V(z)=e^z-1$, in view of our earlier discussion of urns. In words: “a surjection is a sequence of nonempty sets”. See Figure II. 3.1 for an illustration.

## 数学代写|组合学代写Combinatorics代考|Labelled versus unlabelled enumeration

$$\widehat{A}n \leq A_n \leq n ! \widehat{A}_n \quad \text { or equivalently } \quad 1 \leq \frac{A_n}{\widehat{A}_n} \leq n !$$ 示例 5. 标记和末标记的图形。我们在讨论图形时已经遇到过这种现象（图 1)。一般让 $G_n$ 和 $\widehat{G}_n$ 是 size 图的数量 $n$ 分别在标记和末标记的情况下。一个发现 $n=1 \ldots 15$ 序列 \left } { \backslash \text { widehat } { G } { – } n \backslash r i g h t } \text { 构成EIS A000088，可通过第一章方法的扩展得到；参见 [206， } Ch. 4]. 序列 $\backslash$ 左 $\left{G _n \backslash\right.$ 右 $}$ 直接由以下事实决定 $n$ 顶点可以有每个 $(n 2)$ 可能的边缘存在或不存在， 因此
$$G_n=2^{(n 2)}=2^{n(n-1) / 2}$$

## 数学代写|组合学代写Combinatorics代考|Surjections and set partitions

$\mid$ mathcal ${\mathrm{R}}=$ |operatorname ${$ SEQ $} \backslash$ left ${\backslash$ operatorname ${\mathrm{SET}}{\backslash$ geq 1$}{Z} \backslash$ right $}$ |quad $\backslash$ text ${$ and $} \backslash$ quad $\backslash$ math

$$\phi: \quad(2,1,3,4,6,8,9,5,7) .$$

$$\mathcal{R}^{(r)}=\mathrm{SEQ}_r \mathcal{V}, \mathcal{V}=\mathrm{SET} \geq 1 \mathcal{Z} \quad \Longrightarrow \quad R^{(r)}(z)=\left(e^z-1\right)^r$$

## MATLAB代写

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