Question

sa se determine permutarile α ,β ∈ Sn
a) $\frac{ \alpha(1)}{1} = \frac{ \alpha (2)}{2} =...= \frac{ \alpha (n)}{n}$
b)$\frac{ \beta (1)}{n} = \frac{ \beta (2)}{n-1} =...= \frac{ \beta (n)}{1}$

Answer 1

[tex]\frac{\alpha(1)}{1}=\frac{\alpha(2)}{2}=...=\frac{\alpha(n)}{n}=\frac{\alpha(1)+\alpha(2)+...+\alpha(n)}{1+2+...+n}=1, \\ \text{Deoarece }\alpha(1)+\alpha(2)+...+\alpha(n)=1+2+...+n\\ \text{Rezulta asadar, }\\ \alpha(1)=1\cdot1=1,\\ \alpha(2)=2\cdot1=2,\\ ...\\ \alpha(n)=n\cdot1=n\\ \text{Deci }\\ \alpha=\left(\begin{array}{cccc}1&2&...&n\\1&2&...&n\end{array}\right)\\ \text{Pentru $\beta$ se procedeaza analog si se gaseste: }\\ [/tex]
$\beta=\left(\begin{array}{cccc}1&2&...&n\\n&n-1&...&1\end{array}\right)$