some conclusions

Make some conclusions.

 

1. Number of combinations

 $C_n^m = \frac{n!}{m!(n-m)!}$

$C_n^m = C_n^{m-1} + C_{n-1}^{m-1}$

2. Binomial Theorem

$(a+b)^n = \sum\limits_{i=0}^{n}C_n^ia^{n-i}b^i$

3. Burnside Lemma and Polya Theorem

Let $G=\{p_1,p_2,…,p_k\}$ be the permutation group on the target set [1,n]. Then the number of essentially different schemes L:

$L = \frac{1}{|G|}[c(p_1)+c(p_2)+...+c(p_i)]$

$c(p_i)$ represents the number of cycles of length 1 under the permutation $p_i$, that is, the number of fixed points.

 

 

Let G be a permutation group of n objects, and use m colors to color these n objects, then the number of different coloring schemes is:

$L = \frac{1}{|G|}(m^{c(p_1)}+m^{c(p_2)}+...+m^{c(p_k)})$

$c(p_i)$ represents the number of loops to replace $pi$

 

 

 

 

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