Strongly intersecting integer partitions

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