$$S(1\ 2). ( S(12)\ S(1234)\ 1\ S((12)(34))\ S(123))$$

$$= ( S(12)\ S(1234)\ 1\ S((12)(34))\ S(123)) \begin{pmatrix}0&0&1&1&4\\ 0&0&0&2&4\\ 6&0&0&0&0\\ 2&4&0&0&0\\ 3&3&0&0&0 \end{pmatrix}$$

$$A= \begin{pmatrix} 0&0&1&1&4\\ 0&0&0&2&4\\ 6&0&0&0&0\\ 2&4&0&0&0\\ 3&3&0&0&0 \end{pmatrix}$$

�Ȥ����ȡ�$A$ ������¿�༰�ϡ�

$$T^5 - 40 T^3 + 144 T (=T (T - 6) (T - 2) (T + 2) (T + 6))$$

$$A^2= \left(\begin{array}{ccccc} 20 & 16 & 0 & 0 & 0\\ 16 & 20 & ... ... 6 & 6 & 24\\ 0 & 0 & 2 & 10 & 24\\ 0 & 0 & 3 & 9 & 24\\ \end{array}\right)$$

$$A^3= \left(\begin{array}{ccccc} 0 & 0 & 20 & 52 & 144\\ 0 & 0 & ... ... 0 & 0\\ 104 & 112 & 0 & 0 & 0\\ 108 & 108 & 0 & 0 & 0\\ \end{array}\right)$$

$$A^4= \left(\begin{array}{ccccc} 656 & 640 & 0 & 0 & 0\\ 640 & 65... ...64\\ 0 & 0 & 104 & 328 & 864\\ 0 & 0 & 108 & 324 & 864\\ \end{array}\right)$$

$$S(12)^2=6+ 2 S((12)(34))+ 3 S(123)$$

$$S(12)^3=20 S(12)+ 16S(1234)$$

$$S(12)^4=120+ 104S((12)(34))+ 108S(123)$$

$$S(12)^5=656S(12)+ 640S(1234)$$