I know that this kind of transformation by itself without control can lead to contradiction because its value changes depending on the state of the function where you do the transformation. Anyway I want to show my idea:
$$F^k\rightarrow k!$$
Some kind of symbol that it exponent, that it takes on normal exponentiation operations of normal algebra, can be transformed after in a specific moment in an irreversible transformation, i.e., in a function that represents a different thing non-linear as a factorial.
We have, for example, a generating function like
$$f(x)=(1+ax+ax^2+ax^3)(1+ax^3+ax^5)=a^{2} x^{8} + a^{2} x^{7} + 2 \, a^{2} x^{6} + a^{2} x^{5} + a^{2} x^{4} + a x^{5} + 2 \, a x^{3} + a x^{2} + a x + 1$$
then, in some moment, I want to transform the $a^k$ coefficients in $k!$ to express permutations
$$f(x)\rightarrow(a^k\to k!)\to 2! x^{8} + 2! x^{7} + 2\cdot 2! x^{6} + 2! x^{5} + 2! x^{4} + x^{5} + 2 x^{3} + x^{2} + x + 1 $$
I could add another power to express duplicated permutations ($c^k=\frac{1}{k!}$) over groups that I consider equals and lead to repeated patterns.
Is there a symbology or method to do this kind of task?