On nearly self-conjugate partitions of a finite set

Abstract

Let n denote the set of integers 1,2, … , n. Let P =P1, P2, … , Pk be a partition of n. Let C(i) denote the cardinality of the subset Pj to which i belongs. Suppose that P′ = P′1, P′2, … , P′k is a second partition of n and define C′(i) similarly. The partitions P and P′ are called conjugate if (C(i), C′(i)) determine i. If P is a partition of n for which there exists a partition P′ of n such that P and P′ are conjugate and ¦Pi¦ = ¦P′i¦ for all 1 ⩽ i ⩽ k, then P is called nearly self-conjugate. In this paper we prove that for m(m + 1)/2 ⩽ n ⩽ Σ1 ⩽ j ⩽ mj · [m/j], there are nearly self-conjugate partitions of n with max 1 ⩽ i ⩽ k ¦Pi¦ = m, where [x] denotes the greatest integer not exceeding x.