Strongly intersecting integer partitions

Abstract

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.