What is the average value of $\sum\limits_{i=1}^{\pi(k)}\pi(i)$ over all permutaions $\pi$ of $\{1,\ldots, n\}$?

I browsed one enumerative combinatorics problem in Zhihu.

This is also a problem on JAM October 2023.

Proprosed by Erik Vigren. Swedish Institute of Space Physics, Uppsala, Sweden. Suppose that $k$ and $n$ are integers with $n\geq 2$ and $1\leq k\leq n$. What is the average value of $\sum\limits_{i=1}^{\pi(k)}\pi(i)$ over all permutaions $\pi$ of $\{1,\ldots, n\}$?

Below is my Mathematica code:

n = 3; k = 2; kMax = 6; nMax = 6; calc[n_, k_] := Module[{}, permus = Permutations[Range[n]]; Mean[Table[Sum[p[[i]], {i, 1, p[[k]]}], {p, permus}]]] ref[n_, k_] := (n + 1) (3 n + 2)/12 + (n - k + 1) (k - 1)/(2 n - 2) Table[calc[n, k], {n, 2, nMax}, {k, 1, n}] // TableForm[#, TableHeadings -> {Table["n=" <> ToString@i, {i, 2, nMax}], Table["k=" <> ToString@i, {i, 1, kMax}]}] & Table[ref[n, k], {n, 2, nMax}, {k, 1, n}] // TableForm[#, TableHeadings -> {Table["n=" <> ToString@i, {i, 2, nMax}], Table["k=" <> ToString@i, {i, 1, kMax}]}] &