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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.

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    $\begingroup$ You can type almost literally what you have written in the question: Try S[f_][x_,y_]:=y*f[y,x] (see Blank/_ and SetDelayed/:=), and take a look at Nest and NestList $\endgroup$
    – Lukas Lang
    Oct 17, 2022 at 21:35
  • $\begingroup$ Alternatively, S[expr_]:=y*(expr/.{x->y,y->x}); Rest[NestList[S,x^5+y^7,6]]. $\endgroup$
    – user293787
    Oct 17, 2022 at 23:18

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