Can someone explain what the right side means? I'm confused about the notation, especially the {} brackets.

Question

This is asking for a combinatorial proof of $x^n = \sum_{k=0}^n {n \brace k} (x)_k$. I am asking for help interpreting the right side.

Answer 1

By $\{n,k\}$, the authors almost certainly mean the stirling numbers of the second kind which are usually typeset as ${n\brace k}$.

By $x^{\underline{k}}$ the authors mean the falling factorials $x^{\underline{k}} = (x-0)(x-1)(x-2)\ldots(x-(k-1))$.

This fact about stirling numbers is super well known, and basically says that they tell us how to change basis from the standard polynomial basis $x^n$ to the "falling" basis $x^{\underline{n}}$. You can find a proof in lots of places, like here.

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I hope this helps ^_^