

A089299


Number of square plane partitions of n.


13



1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 41, 57, 78, 108, 146, 202, 274, 375, 509, 690, 929, 1255, 1679, 2246, 2991, 3979, 5266, 6971, 9187, 12104, 15898, 20870, 27322, 35762, 46690, 60927, 79348, 103270, 134138, 174108, 225576, 291990, 377320, 487083
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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..2k1} (1x^j)^min(j,2kj).  Franklin T. AdamsWatters, 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

Cf. A008763, A001970, A089292.
Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.
Sequence in context: A241545 A157162 A109434 * A017910 A240734 A328460
Adjacent sequences: A089296 A089297 A089298 * A089300 A089301 A089302


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. AdamsWatters, Jun 14 2006
Name edited by Gus Wiseman, Jan 16 2019


STATUS

approved



