# 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.