Number of partitions of n = p(n)
Number of partitions of n which has a part equal to 1 = p(n-1)
Number of partitions of n into k parts = p(n,k)
If for some k the following inequality holds
p(n,k) ≤ p(n-1)
Then does it necessarily imply that all the partitions of n into k parts has a part equal to 1?
No, for example $p(4,2) = 2 = p(3)$ but there is a partition of $4$ into $2$ parts where neither part is $1$.