I want to be able to build an operator $S$, on a polynomial in multiple variables which does something like this
$f[x,y] =$ some polynomial
$S[f][x,y] = y*f[y,x]$
And then for a specific choice of $n$, I would like to see the results of $S^k[f]$ for all $1 \leq k \leq n$.
I am very new to Mathematica, practically a baby, so I would appreciate as syntactically simple an explanation as possible.