Example 1 For $a, b, c \in \mathbf{R}^{+}$, compare $a^{3}+b^{3}+c^{3}$ with $a^{2} b+b^{2} c+c^{2} a$.
To apply the "rearrangement inequality," two sets of numbers that can be easily sorted must be selected, which requires analysis of the structure of the two expressions.

Solution: Take two sets of numbers $a, b, c ; a^{2}, b^{2}, c^{2}$. Regardless of the order of $a, b, c$, $a^{3}+b^{3}+c^{3}$ is always a similarly ordered sum, and $a^{2} b+b^{2} c+c^{2} a$ is always a disordered sum. Therefore, we have
$$
a^{3}+b^{3}+c^{3}>a^{2} b+b^{2} c+c^{2} a \text {. }
$$