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.
Grid Shading Problem
See alsoq-Binomial Coefficient, Unimodal Sequence
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ReferencesProctor, R. A. "Solution of Two Difficult Combinatorial Problems with Linear Algebra." Amer. Math. Monthly 89, 721-734, 1982.Sylvester, J. J. "Proof of the Hitherto Undemonstrated Fundamental Theorem of Invariants." Philos. Mag. 5, 178-188, 1878.
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Cite this as:
Weisstein, Eric W. "Grid Shading Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GridShadingProblem.html