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%I #23 Jan 12 2022 09:50:25
%S 1,1,1,3,4,6,10,15,22,33,49,70,102,146,205,290,405,561,779,1071,1463,
%T 1999,2714,3667,4946,6641,8880,11848,15753,20870,27586,36354,47766,
%U 62621,81878,106785,138975,180449,233778,302270,390027,502256,645603,828330,1060851
%N Number of planar partitions of n where parts strictly decrease along each row and column.
%C 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.
%D B. Gordon, Multirowed partitions with strict decrease along columns (Notes on plane partitions IV.), Symposia Amer. Math. Soc. 19 (1971) 91-100.
%H Alois P. Heinz, <a href="/A114736/b114736.txt">Table of n, a(n) for n = 0..85</a>
%e For n = 5, we have the 6 partitions [5], [4,1], [4|1], [3,2], [3|2] and [3,1|1].
%e From _Gus Wiseman_, Nov 15 2018: (Start)
%e The a(6) = 10 plane partitions:
%e 6 5 1 4 2 3 2 1
%e .
%e 5 4 1 4 3 2 3 1
%e 1 1 2 1 2
%e .
%e 3
%e 2
%e 1
%e (End)
%t prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t 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 *)
%Y Cf. A000009, A000219, A001970, A007716, A068313, A117433, A120733, A319646, A321645, A321652, A321653, A321655.
%K nonn
%O 0,4
%A _Franklin T. Adams-Watters_, Mar 16 2006
%E Clarified definition, added 30 terms and reference. - _Dennis K Moore_, Jan 12 2011
%E a(40)-a(44) from _Alois P. Heinz_, Sep 26 2018
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