Intersecting integer partitions

Abstract

If a1, a2, . . . , ak and n are positive integers such that n = a1+a2+· · ·+ak, then the sum a1 + a2 + · · · + ak is said to be a partition of n of length k, and a1, a2, . . . , ak are said to be the parts of the partition. Two partitions that differ only in the order of their parts are considered to be the same partition. Let Pn be the set of partitions of n, and let Pn,k be the set of partitions of n of length k. We say that two partitions t-intersect if they have at least t common parts (not necessarily distinct). We call a set A of partitions t-intersecting if every two partitions in A t-intersect. For a set A of partitions, let A(t) be the set of partitions in A that have at least t parts equal to 1. We conjecture that for n ≥ t, Pn(t) is a largest t-intersecting subset of Pn. We show that for k > t, Pn,k(t) is a largest t-intersecting subset of Pn,k if n ≤ 2k − t + 1 or n ≥ 3tk5. We also demonstrate that for every t ≥ 1, there exist n and k such that t < k < n and Pn,k(t) is not a largest t-intersecting subset of Pn,k.