On nearly self-conjugate partitions of a finite set

✍ Scribed by Honghui Wan

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 374 KB
• Volume: 175
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(94)00066-r

✦ Synopsis

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'~, P~ .... , P~,} 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' ofn such that P and P' are conjugate and tPi] = ]PI] for all 1 ~< i ~< k, then P is called nearly self-conjugate. In this paper we prove that for rn(m + 1)/2 <~ n <. ~1 ~.j~m J" [m/j],

there are nearly self-conjugate partitions of n with max1 ~iSk IP~l = rn, where Ix] denotes the greatest integer not exceeding x.