Perfect partitions (Q1207173)

A partition of \(n\) is said to be perfect if it contains as a subset exactly one partition of every integer up to \(n\). For any integer \(n\), the partition consisting of \(n\) 1's is perfect and for odd \(n=2k+1\), the partition consisting of one 1 and \(k\) 2's is perfect. These are called trivial perfect partitions. The author determines those \(n\) for which there exist nontrivial perfect partitions and the number of perfect \(n\)-partitions, denoted \(\text{Per}(n)\). He shows that the \(n\)-partition \((a_ 1,a_ 2,\dots,a_ s)\) is perfect if and only if \(a_ 1=1\) and \(a_ t=a_{t-1}\) or \(a_ t=a_ 1+a_ 2+\cdots+a_{t-1}+1\) for \(1<t\leq s\). He also shows that there exist nontrivial partitions for \(n\) if and only if \(n+1\) is not a prime. He includes a table of values for \(\text{Per}(n)\) for all \(n\) from 1 to 100 and a table of values for \(\text{Per}(n)\) in terms of the prime factorization of \(n+1\). For example, if \(n+1\) has the form \(q^ 2_ 1 q^ 2_ 2\) then \(\text{Per}(n)=26\), while if \(n+1=q^ 3_ 1 q^ 2_ 2 q_ 3\), then \(\text{Per}(n)=604\).