Are values of multinomials distinct for distinct sets of integer partitions in the denominator?

Question

Let a multinomial be denoted by $$M(n, K) = {n! \over {\prod k_j!}}$$ where $K= (k_1, k_2, ..., k_n)$ and $k_1 \ge k_2 \ge ... \ge k_n$.

It is obvious that K is an integer partition of n. Then, my question is:

Is $M(n, K_1) \ne M(n, K_2)$ if $K_1\ne K_2$. In other words, is $M(n, K)$ unique for a given $n$?

Answer 1

Note that $\binom{16}{2}=\binom{10}{3}$. Therefore $10!2!14!=16!3!7!$. Consider $K_1=\{14,10,2\}$ and $K_2=\{16,7,3\}$ and then we have $M(26,K_1)=M(26,K_2)$ for $K_1\neq K_2$.