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# 数学代写|组合学代写Combinatorics代考|CS-E4555 Burnside’s Lemma

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## 数学代写|组合学代写Combinatorics代考|Burnside’s Lemma

The simple problem of coloring fields of a square $2 \times 2$ using three colors is considered in Example 2.7.11, and the number of nonequivalent colorings is determined by direct counting. At the beginning of this section we shall formulate two more similar problems.

Example 7.1.1. Coloring the vertices of a cube using blue, yellow, and red colors can be done in $3^{8}=6561$ ways. Obviously, we are considering here the case when not all three colors are necessarily used. Two colorings are equivalent (or geometrically equal) if there is a rotation $\mathcal{R}$ of the cube such that the image $\mathcal{R}(v)$ of any vertex $v$ is of the same color as the vertex itself. Let us consider some classes of colorings.

Class $\mathcal{A}$ consists of colorings such that 4 vertices that belong to the same side of the cube are red, and the remaining 4 vertices are blue. All colorings from class $\mathcal{A}$ are obviously equivalent. Class $\mathcal{B}$ consists of colorings such that three vertices are blue, and the remaining 5 vertices are red. Class $\mathcal{C}$ consists of colorings such that each of the three colors is assigned to at least one vertex.

## 数学代写|组合学代写Combinatorics代考|On Permutations

In this section we shall introduce some important notions related to permutations of the set $\mathbb{N}{n}={1,2, \ldots, n}$. We start with the notions of inversion and the parity (oddness and evenness) of a permutation. An arbitrary permutation of set $\mathbb{N}{n}$ will be denoted by $a_{1} a_{2}, \ldots a_{n}$.

Inversion. Consider elements $i, j \in \mathbb{N}{n}$. We say that the pair $(i, j)$ is an inversion of a permutation $a{1} a_{2} \ldots a_{n}$ if $ia_{j}$. For example, let us consider the permutation 3142 of the set ${1,2,3,4}$. All the inversions of this permutation are $(3,1),(3,2)$, and $(4,2)$.

Parity of a permutation. A permutation $a_{1} a_{2} \ldots a_{n}$ is odd if the number of all its inversions is odd. A permutation $a_{1} a_{2} \ldots a_{n}$ is even if the number of all its inversions is even.

Theorem 7.2.1. Consider a permutation $a_{1} a_{2} \ldots a_{n}$. If two elements $a_{i}$ and $a_{j}$ exchange positions, then the permutation changes parity.

Proof. Suppose there are exactly $k$ elements between $a$ and $b$ in the permutation $a_{1} a_{2} \ldots a_{n}$, where $a, b \in\left{a_{1}, a_{2}, \ldots, a_{n}\right}$. This permutation has the following form
$$a_{1} a_{2} \ldots a c_{1} c_{2} \ldots c_{k} b \ldots a_{n} .$$
Elements $a$ and $b$ can exchange positions the following way. First, element $a$ exchanges positions with elements $c_{1}, c_{2}, \ldots, c_{k}$ and $b$, one after the other, and then element $b$ exchanges positions with elements $c_{k}, c_{k-1}, \ldots, c_{1}$, one after the other. This way, exactly $2 k+1$ exchanges of the positions of neighboring elements were made. Note that every exchange of the positions of neighboring elements changes the parity of the permutation. Consequently, after $2 k+1$ steps, the obtained permutation is not of the same parity as the starting one, and the proof is completed.

## 数学代写|组合学代写Combinatorics代考 $\mid$ On Permutations

$$a_{1} a_{2} \ldots a c_{1} c_{2} \ldots c_{k} b \ldots a_{n}$$

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