TITLE: Partitions to different  parts not exceeding $n$
QUESTION [25 upvotes]: 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.

REPLY [34 votes]: 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.