Ring of symmetric functions

Okay, imagine you have some toys, like building blocks or dolls. And each toy has some colors and shapes.

Now, imagine you want to sort all of your toys based on their colors and shapes. And you want to make sure that if you switch the colors and shapes of two toys, the order of the sorted toys doesn't change.

This is kind of like what we do with symmetric functions. We take different variables, like x, y, and z, and we put them in different combinations and powers to make functions. Then, we sort these functions based on their symmetry.

Symmetry means that if you swap two variables or change the order of them, the function looks the same. Like how your toys would still be sorted even if you switched their colors and shapes around.

The Ring of Symmetric Functions has a lot of different functions that have different symmetries. We use them in math to study things like polynomials and equations. It's kind of like a big toy box filled with different toys, where we can play with them and see how they all fit together.