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A114736
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Number of planar partitions of n where parts strictly decrease along each row and column.
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27
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1, 1, 1, 3, 4, 6, 10, 15, 22, 33, 49, 70, 102, 146, 205, 290, 405, 561, 779, 1071, 1463, 1999, 2714, 3667, 4946, 6641, 8880, 11848, 15753, 20870, 27586, 36354, 47766, 62621, 81878, 106785, 138975, 180449, 233778, 302270, 390027, 502256, 645603, 828330, 1060851
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OFFSET
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0,4
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COMMENTS
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If these partitions are "flattened" into a simple partition, the resulting partitions are those for which any part size present with multiplicity k implies the presence of at least k(k-1)/2 larger parts. E.g., [3,1|1] flattens to [3,1^2], 1 has multiplicity 2, so there must be at least 2*1/2 = 1 part larger than 1 - which is the 3.
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REFERENCES
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B. Gordon, Multirowed partitions with strict decrease along columns (Notes on plane partitions IV.), Symposia Amer. Math. Soc. 19 (1971) 91-100.
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LINKS
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EXAMPLE
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For n = 5, we have the 6 partitions [5], [4,1], [4|1], [3,2], [3|2] and [3,1|1].
The a(6) = 10 plane partitions:
6 5 1 4 2 3 2 1
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5 4 1 4 3 2 3 1
1 1 2 1 2
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3
2
1
(End)
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MATHEMATICA
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prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], And@@(OrderedQ[#, Greater]&/@prs2mat[#]), And@@(OrderedQ[#, Greater]&/@Transpose[prs2mat[#]])]&]], {n, 5}] (* Gus Wiseman, Nov 15 2018 *)
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CROSSREFS
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Cf. A000009, A000219, A001970, A007716, A068313, A117433, A120733, A319646, A321645, A321652, A321653, A321655.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Clarified definition, added 30 terms and reference. - Dennis K Moore, Jan 12 2011
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STATUS
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approved
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