OFFSET
0,5
COMMENTS
Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..4000
FORMULA
G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - Franklin T. Adams-Watters, Jun 14 2006
EXAMPLE
a(7) = 5:
7 41 32 31 22
. 11 11 21 21
a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}
From Gus Wiseman, Jan 16 2019: (Start)
The a(10) = 16 square plane partitions:
[ten]
.
[32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]
[32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]
.
[211]
[111]
[111]
(End)
MATHEMATICA
Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#, {Length[ptn]}]&/@ptn]], And@@OrderedQ/@Transpose[#]&]], {ptn, IntegerPartitions[n]}], {n, 30}] (* Gus Wiseman, Jan 16 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 25 2003
EXTENSIONS
Corrected and extended by Wouter Meeussen, Dec 30 2003
a(21)-a(25) from John W. Layman, Jan 02 2004
More terms from Franklin T. Adams-Watters, Jun 14 2006
Name edited by Gus Wiseman, Jan 16 2019
STATUS
approved