Working on a problem in the symmetric group I have stumbled upon the following equation:

$$sum_{substack{pi=(1^{c_1},2^{c_2},ldots,n^{c_n})\textrm{partition of }n}}(-1)^{n-sum_{i=1}^nc_i}frac{n!}{prod_{i=1}^ni^{c_i}c_i!}left(sum_{substack{eta=(1^{b_1},2^{b_2},ldots,k^{b_k})\textrm{partition of }k}}prod_{j=1}^k{c_jchoose b_j}right)^ell=0,$$

where $n,k$ and $ell$ are positive integers and $1le k,ellle n-1.$ (Here, $pi=(1^{c_1},2^{c_2},ldots,n^{c_n})$ means that $n=1cdot c_1+2cdot c_2+cdots +c_ncdot n$.)

I have two questions. First, whether this equality has appeared somewhere. (I have checked the book(s) of Stanley on Enumerative combinatorics, but with no luck.)

Second, given $n$ and $k$, I am intersted on the smallest value of $ell$, where the equality above is not satisfied. For instance, when $n:=43$ and $k:=13$, with a computer computation one can check that the equality above is satisfied FOR EACH element in ${1,2,3,4,5,6}$ and $ell:=7$ is the first time where this equality is not satisfied. However, I have no clue in how to get my hands on this value!