Hi Mr.

let a = beta(x), b=beta(y).

Then a neq b and 1\le a ,b \le n.

If a < b then beta is in Z. If a < b then beta is in W. Thus

W and Z partition S_n.

Note that beta alpha(x) = beta(y) = b

and beta alpha(y) = beta(x) = a.

Thus,

beta in W

iff

beta(x) > beta(y)

iff

a > b

iff

beta alpha(y) = a > b = beta alpha(x)

\iff

beta alpha in Z

Thus, the map f that takes beta to beta alpha sends

everything in W to Z.

Since f applied twice is the identity, it follows that f also sends

everything in Z back to W.

I.e., f is a bijection between W and Z.

Best,

ZD