Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or maybe this is false? 


This problem was proposed long ago on some Russian high school competition, but nobody managed to solve, including jury.



Partitions to different  parts not exceeding $n$
In fact, you *cannot* prove unimodality of the coefficients of $P\_n(x)=(1+x)\cdots (1+x^n)$ using the result about $B\_m/G$ mentioned by Qiaochu. The $P\_n(x)$ result is implicit in work of Dynkin on the principal sl(2) subalgebra of a complex semisimple Lie algebra. Hughes was the first to realize that a special case of Dynkin's result implied the unimodality of $P\_n(x)$. Proctor removed all the Lie algebra theory from the proof, yielding a proof involving only elementary linear algebra. See <http://math.mit.edu/~rstan/pubs/pubfiles/72.pdf>, esp. equation (23). See also <http://math.mit.edu/~rstan/algcomb/algcomb.pdf>, beginning on page 90. It is a long-standing open problem to find a combinatorial proof of this unimodality result. I would be extremely impressed if a high school student could prove unimodality by any method.


