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A117433 Number of planar partitions of n with all part sizes distinct. 26
1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 149, 187, 243, 321, 413, 527, 735, 895, 1165, 1467, 1885, 2335, 2997, 3853, 4765, 5977, 7473, 9269, 11531, 14255, 17537, 22201, 26897, 33233, 40613, 50027, 60637, 74459, 89963, 109751, 134407, 162117, 195859 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Matches A072706 for n < 10, since a unimodal composition into distinct parts can be placed uniquely as a hook. Starting with n = 10, additional partitions are possible (starting with [4,3|2,1] and [4,2|3,1]).

LINKS

Franklin T. Adams-Watters and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Franklin T. Adams-Watters)

OEIS Wiki, Plane partitions

FORMULA

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} A000085(k)*A008289(n,k).

EXAMPLE

From Gus Wiseman, Nov 15 2018: (Start)

The a(10) = 35 strict plane partitions (A = 10):

  A  64  73  82  532  91  541  631  721  4321

.

  9  54  63  72  432  8  53  71  431  7  43  52  61  421  6  42  51

  1  1   1   1   1    2  2   2   2    3  21  3   3   3    4  31  4

.

  7  6  5  43  42  5  41

  2  3  4  2   3   3  3

  1  1  1  1   1   2  2

.

  4

  3

  2

  1

(End)

MAPLE

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)

      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))

    end:

g:= proc(n) g(n):= `if`(n<2, 1, (n-1)*g(n-2) +g(n-1)) end:

a:= proc(n) b(n, n); add(%[i]*g(i-1), i=1..nops(%)) end:

seq(a(n), n=0..60);  # Alois P. Heinz, Nov 18 2012

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/@#], UnsameQ@@DeleteCases[Join@@prs2mat[#], 0], And@@(OrderedQ[#, Greater]&/@prs2mat[#]), And@@(OrderedQ[#, Greater]&/@Transpose[prs2mat[#]])]&]], {n, 5}] (* Gus Wiseman, Nov 15 2018 *)

CROSSREFS

Cf. A000219, A072706, A117434, A000009.

Cf. A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321652, A321653, A321655, A321659, A321660.

Sequence in context: A102437 A319794 A072706 * A159284 A078028 A317142

Adjacent sequences:  A117430 A117431 A117432 * A117434 A117435 A117436

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Mar 16 2006, Apr 01 2008

STATUS

approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)