Strongly intersecting integer partitions

We call a sum a1+a2+⋯+ak a partition of n of length k if a1,a2,…,ak and n are positive integers such that a1≤a2≤⋯≤ak and n=a1+a2+⋯+ak. For i=1,2,…,k, we call ai the ith part of the sum a1+a2+⋯+ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+⋯+ak and b1+b2+⋯+bk strongly intersect if ai=bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2≤k≤n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k≥4 or k=3≤n∉{6,7,8} or k=2≤n≤3.