  We study M(n), the number of distinct values taken by multinomial
coefficients with upper entry n, and some closely related sequences. We show
that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where
pP(n) is the number of partitions of n into primes and p(n) is the total number
of partitions of n. To use methods from commutative algebra, we encode
partitions and multinomial coefficients as monomials.
