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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 (list; graph; refs; listen; history; text; internal format)
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

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. Adams-Watters, Jun 14 2006

Name edited by Gus Wiseman, Jan 16 2019

STATUS

approved

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Last modified May 25 13:00 EDT 2022. Contains 354071 sequences. (Running on oeis4.)