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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)

PROG

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 June 25 14:30 EDT 2019. Contains 324352 sequences. (Running on oeis4.)