The grid shading problem is the problem of proving the unimodality of the sequence ,
where for fixed
and ,
is the number of partitions of with at most parts and largest part at most . The grid shading problem was solved by Sylvester (1878) using
invariant theory (Proctor 1982). Proctor (1982) gave the first elementary proof of
this result.

Proctor, R. A. "Solution of Two Difficult Combinatorial Problems with Linear Algebra." Amer. Math. Monthly89, 721-734,
1982.Sylvester, J. J. "Proof of the Hitherto Undemonstrated
Fundamental Theorem of Invariants." Philos. Mag.5, 178-188, 1878.