TR-MO 1993 Problem 3

$n\in{Z^{+}}$ and $A={1,\ldots ,n}$ . $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$ . $S_\sigma$ shows the set of good functions for $\sigma$ . a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation. b) $n\geq 4$ , prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$ .