Grid Shading Problem

The grid shading problem is the problem of proving the unimodality of the sequence {a_1,a_2,...,a_(mn)}, where for fixed m and n, a_i is the number of partitions of i with at most m parts and largest part at most n. The grid shading problem was solved by Sylvester (1878) using invariant theory (Proctor 1982). Proctor (1982) gave the first elementary proof of this result.

The q-binomial coefficients give the generating function for this sequence.

See also

q-Binomial Coefficient, Unimodal Sequence

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Proctor, 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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Grid Shading Problem

Cite this as:

Weisstein, Eric W. "Grid Shading Problem." From MathWorld--A Wolfram Web Resource.

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