On graphical partitions (Q2367442)

For even \(n\), let \(p(n)\) denote the number of partitions of \(n\) and \(G(n)\) denote the number of graphical partitions of \(n\). A partition \(\pi=(\lambda_1,\lambda_2,\dots,\lambda_m)\) is graphical if there exists a graph with degree sequence \(\pi\). The authors discuss progress and possible lines in enquiry on the questions of whether or not \(\lim_{n\to\infty}G(n)/p(n)\) approaches 0, and prove two inequalities: \[ \limsup_{n\to\infty}{G(n)\over P(n)}\leq .4258,\;\liminf_{n\to\infty}n^{1/2}{G(n)\over P(n)}\geq{\pi\over\sqrt 6}. \]