Partitions into odd, unequal parts
Let \(q(j,n)\) denote the number of partitions of \(j\geq 2\) into odd unequal parts none exceeding \(2n-1.\) Then the author and R. P. Stanley conjecture that \(q(j,n)\) increases with j for \(2\leq j\leq n^ 2/2\). Here is the reviewer's proof of the conjecture. Let (1) \(j=u_ 1,u_ 2,u_ 3,\) \(u_ 4,...,u_ k,\) where the u's are distinct odd numbers arranged in ascending order of magnitude, be a partition of j into distinct odd parts. If 1 appears as a part on the right of (1), then (2) \(j+1=1\), \(u_ 1,u_ 2,u_ 3,\) \(u_ 4,...,u_ k\) provides a partition of \((j+1)\) which is not a partition of j. If 1 does not appear as a part on the right of (1) then \(j+2=u_ 1,u_ 2,u_ 3,\) \(u_ 4,...,(u_ k+2)\) provides a partition of \(j+2\) into distinct odd parts which is different from the partition of \((j+1)\) in (2) and also from the partition of j in (1). This proves the conjecture of the author and Stanley.
scientific article; zbMATH DE number 3935044