symmetric groups – How to prove this identity on summations and partitions?

Let $ f $ to be a symmetrical function of $ s $ variables. The identity is
$$ sum_ {all k}} ^ infty x ^ { sum_ {j = 1} ^ kk}} (k_1, k_2, k_3, …, k_s) = sum_ {n = s} ^ infty x ^ n sum _ { lambda vdash n} frac {s! prod_l lambda_l} {z_ lambda} f ( lambda) $$
or $ z_ lambda $ is the size of the centering device of a type permutation $ lambda $. As you can see $ frac {s! prod_l lambda_l} {z_ lambda} $ is the number of compositions with the elements of $ lambda $. And do not forget that $ lambda $ is always a partition with $ s $ rooms.

I checked it for some values ​​of $ s $. For example (s = 3):

$$
sum_ {k_1 = 1} ^ infty sum_ {k_1 = 1} ^ infty sum_ {k_1 = 1} ^ infty x 3f (1,1,1) + 3x ^ 4f (2,1,1) + x ^ 5[3f(2,2,1)+3f(3,1,1)]+ x ^ 6[f(2,2,2)+6f(3,2,1)+3f(4,1,1)]…
$$

It is obvious that the model is emerging.

How can this be rigorously proven? (for everyone $ s in mathbb {N} $ of course!) A reference on this kind of manipulation?

Also, if you have a better way to write the number $ frac {s! prod_l lambda_l} {z_ lambda} $ it will be useful.