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 A114736 Number of planar partitions of n where parts strictly decrease along each row and column. 27
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. REFERENCES B. Gordon, Multirowed partitions with strict decrease along columns (Notes on plane partitions IV.), Symposia Amer. Math. Soc. 19 (1971) 91-100. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..85 EXAMPLE For n = 5, we have the 6 partitions [5], [4,1], [4|1], [3,2], [3|2] and [3,1|1]. From Gus Wiseman, Nov 15 2018: (Start) The a(6) = 10 plane partitions: 6 5 1 4 2 3 2 1 . 5 4 1 4 3 2 3 1 1 1 2 1 2 . 3 2 1 (End) MATHEMATICA 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 *) CROSSREFS Cf. A000009, A000219, A001970, A007716, A068313, A117433, A120733, A319646, A321645, A321652, A321653, A321655. Sequence in context: A171096 A125869 A059618 * A099417 A139463 A287067 Adjacent sequences: A114733 A114734 A114735 * A114737 A114738 A114739 KEYWORD nonn AUTHOR Franklin T. Adams-Watters, Mar 16 2006 EXTENSIONS Clarified definition, added 30 terms and reference. - Dennis K Moore, Jan 12 2011 a(40)-a(44) from Alois P. Heinz, Sep 26 2018 STATUS approved

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Last modified October 2 21:26 EDT 2023. Contains 365840 sequences. (Running on oeis4.)