R.P. Stanley's and G-N. Han's work in 2008-2009 on Hook length formulas do not seem to connect sums over partitions of different sizes. I mean relations between sums over partitions of n-1, of n and of n+1. I got one, but do not know if I should be surprised or not. It takes a better mathematician to distinguish between the boring and the interesting tidbits.
Since I'm a hopeless dud at LaTeX, forgive me replacing it with a link to a PDF file with the accompanying short Mathematica 4.0 notebook: