Let $X = \{x_1,x_2,\dots,x_n\}$ and $Y = \{y_1,y_2,...,y_n\}$ and let $f:X\to Y$ is it acceptable to represent this function using the following notation or is it reserved only for permutations.
$$\begin{pmatrix}x_1&x_2&\dots&x_n\\f(x_1)&f(x_2)&\dots&f(x_n)\end{pmatrix}$$
It is a standard notation in semigroup theory to denote transformations. See for instance
O. Ganyushkin and V. Mazorchuk, Classical finite transformation semigroups. An introduction. Algebra and Applications, 9. Springer-Verlag London, Ltd., London, 2009. xii+314 pp. ISBN: 978-1-84800-280-7