If N is a finite list of numbers, let $p(N)$ be the product of the numbers in N, that is, $$p(N)=\Pi_{i=1}^{|N|}N_i$$ and let $s(N)$ be the sum of the partial products of the numbers in N, that is $$s(N)=\Sigma_{k=1}^{|N|}\left(\Pi_{i=1}^kN_i\right)$$
Is there a pair of lists $N\neq M$ for which $p(N)=p(M)$ and $s(N)=s(M)$?
What if they only contain odd numbers 3 or higher?
What if $N$ and $M$ only contain prime numbers? Only odd primes?
What if they only contain elements of $\{3,7,19\}$? (This is the case I'm most interested in)
It seems possible that there's no such (distinct) $N$ and $M$ for the last case, but I've no idea how to prove that.