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Descending Plane Partition


 7 7 6 6 3 1;  6 5 4 2 ;   3 3  ;    2

A descending plane partition of order n is a two-dimensional array (possibly empty) of positive integers less than or equal to n such that the left-hand edges are successively indented, rows are nonincreasing across, columns are decreasing downwards, and the number of entries in each row is strictly less than the largest entry in that row. Implicit in this definition are the requirements that no "holes" are allowed in the array, all rows are flush against the top, and the diagonal element must be filled if any element of its row is filled. The above example shows a decreasing plane partition of order seven.

 3 3;  2  3 3  3 2  3 1  3  2  emptyset

The sole descending plane partition of order one is the empty one emptyset, the two of order two are "2" and emptyset, and the seven of order three are illustrated above. In general, the number of descending plane partitions of order n is equal to the number of +1-bordered alternating sign matrices: 1, 2, 7, 42, 429, ... (OEIS A005130).


See also

Alternating Sign Matrix, Plane Partition

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References

Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math. 53, 193-225, 1979.Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637-646.Sloane, N. J. A. Sequence A005130/M1808 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Descending Plane Partition

Cite this as:

Weisstein, Eric W. "Descending Plane Partition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DescendingPlanePartition.html

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