A114736 Number of planar partitions of n where parts strictly decrease along each row and column.

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

Offset: 0

Author

Franklin T. Adams-Watters, Mar 16 2006

Keywords

• nonn

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.

Examples

			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)
		

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

Crossrefs

Cf. A000009, A000219, A001970, A007716, A068313, A117433, A120733, A319646, A321645, A321652, A321653, A321655.

Programs

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 *)

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