Two polynomials P(x) and Q(x) commute whenever P(Q(x)) = Q(P(x)).

Some polynomials commute, and others don't. The set of powers $$\{x^1, x^2, x^3, \ldots, \}$$ is an example of a chain—a list of polynomials, one of each degree, which all commute with each other. Are there any other chains?

Any linear polynomial $$\lambda(x)=ax+b$$, (where a is nonzero), has an inverse $$\lambda^{-1}(y)=(y-b)/a$$. And if $$\{P_i(x)\}$$ is a chain, then $$\{\lambda^{-1}\circ P_i \circ \lambda\}$$ is also a chain. So every chain is part of a family of similar polynomials. Up to similarity, there is only one other polynomial chain: the Chebyshev polynomials $$T_n(x) \equiv \cos(n\cdot \cos^{-1}(x))$$.

Why? You can prove that any quadratic polynomials commutes with at most one polynomial of each degree (so the quadratic determines the whole chain), and each quadratic is similar to a quadratic of the form $$x^2+c$$. In chains, the commuting constraint forces $$c(c+2)=0$$. When c=0, you get the power polynomials. When c=-2, you get the Chebyshev polynomials.