I'm trying to generate to monomial symmetric polynomial $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 + \text{all permutations,} $$ starting from the first element by applying all necessary permutations.
Since $m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})$ has $\frac{10!}{6!}=5040$ elements, I think that not all elements of $S_{10}$ are necessary and I wonder if there is a subgroup of it.
I found $S_7$ to have the right number of elements, but first this doesn't make sense to me and second my next example would be $m_{(1,2,3,4,5)}(X_1,X_2,\dots,X_{15})$, where $S_9$ doesn't match (see here for a related question).