I. Definition
- group
Group is what? ? ? I will not ah
- Replacement ( \ (G \) )
A replacement is an operation on behalf of a way to make objects change places
- Permutation group ( \ (G \) )
As the name suggests, the group formed by the permutation
- Fixed displacement class k ( \ (Z_K \) ) (stabilized facilitator)
That the element \ (K \) set without changing the position of the group
- Equivalence class ( \ (E_k \) ) (track)
Permutation group \ (G \) under the action of an element \ (K \) trajectory (collection of points)
- Loop ( \ (h_g \) )
In substitution \ (G \) cycles under the action generated
- Track - Polar stable Theorem
\ [| E_k | \ times |
Z_k | = | G | \] proof: No
- burnside lemma
\ [L = \ frac {1} {| G |} \ sum c_i (c_i constant indicates the number of elements in replacement I) \]
By the track - Polar apparent stability theorems, | G | an equivalence class may represent all elements \ (Z_K \) sum
There \ [L \ times | G | = \ sum_ {i = 1} ^ n | Z_i | \]
According to the definition, we have \ [\ sum_ {i = 1 } ^ n | Z_i | = \ sum_ {k = 1} ^ {| G |} c_i \]
则\[L=\frac{1}{|G|}\sum c_i\]
- polya Theorem
\ [L = \ frac {1} {| G |} \ sum_ {i = 1} ^ {| G |} m ^ {h_i} (m is the number of colors) \]
No location restrictions only apply to the case of color
Can apparently found in all colors equal conditions and \ (burnside \) Lemma is the same as
two. Example
- Questions 1. Most of permutation groups are set with \ (burnside \) skin \ (dp \) , without repeat them here
[Bzoj1851] color colored FIG.
Description meaning of problems: a complete graph of n nodes, to color with edge coloring m, for switching point numbers isomorphic, asked how many different staining protocol
Charme good friends speak of God group theory of the blog