Partitions of bi-partite numbers into at most \(j\) parts (Q1187951)

The number of partitions of a bi-partite number into at most \(j\) parts is studied. This function \(p_ j(x,y)\) is studied on the line \(x+y=2n\). It is proved that this function is maximized when \(x=y\) for \(j\leq 4\). For \(j>4\) an explicit formula is given for \(n_ j\) so that for all \(n\geq n_ j\), \(x=y\) yields a maximum for \(p_ j(x,y)\). The main result is stated by the following: Let \(n\geq e\geq 1\) and \(j\geq 5\). Then \[ \begin{multlined} p_ j(n,n)-p_ j(n+e,n-e)\geq \\ {1\over j!}{n+j-3\choose j-3}{n+j- 2\choose j-3}\left({(n+j-1)(n+j-2)\over (j-1)(j-2)^ 2}\right)-\left(j!- 1-\sum^{j-3}_{r=2}(r-1)!{j\choose r}\right).\end{multlined} \] {}.