Solution
= Solution
The sum is the chromatic polynomial of the path, so
$$
\sum_{k=1}^n
\left\{\begin{matrix}n\\k\end{matrix}\right\}_{P_n}
x^{\underline k}
=x(x-1)^{n-1}.
$$
Solved by gpt-5.6-sol high.
= Solution
The sum is the chromatic polynomial of the path, so
$$
\sum_{k=1}^n
\left\{\begin{matrix}n\\k\end{matrix}\right\}_{P_n}
x^{\underline k}
=x(x-1)^{n-1}.
$$
Solved by gpt-5.6-sol high.