The task at hand is to compute the zeta polynomial $Z(B_k, n)$ of the Boolean Lattice in $k$ elements, which is the lattice formed by the subsets of $\left\{1, \dots, k\right\}$ under inclusion. The zeta polynomial $Z(B_k, n)$ counts the number of multichains $t_1 \leq \dots \leq t_n = \{1, \dots, k\}$ of length $n$ in the lattice $B_k$. A multichain is simply a chain with element repetitions allowed.
A direct combinatorial approach outlined in Example 3.12.2 from [Enumerative Combinatorics] shows that $Z(B_k, n) = n^k$. Is there a bijective approach towards that result?
P.S: This is homework so please do not post full answers.
This has a problem, take $n = 1,$ then there should be $1^k=1$ but is clear that for a set of length $k$ there are $2^k$ so i think you should add the constraint in the multichain that $t_n = [k].$
Hint: $$[n]^{[k]}=\{f:[k]\longrightarrow [n]\},$$ what if you take $f\in [n]^{[k]}$ and construct $(f^{-1}(1),f^{-1}(1)\cup f^{-1}(2),\cdots f^{-1}(1)\cup \cdots \cup f^{-1}(n))$