A003293 -id:A003293 - cp's OEIS frontend

A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305

Offset: 0

David S. Newman, Nov 21 2004

nonn

From Gus Wiseman, Jan 21 2019: (Start)
Also the number of semistandard Young tableaux where the rows are constant and the entries sum to n. For example, the a(8) = 19 tableaux are:
  8   44   2222   11111111
.
  1   2   11   3   111   22   1111   11   11111   1111   111111
  7   6   6    5   5     4    4      33   3       22     2
.
  1   1   11   111
  2   3   2    2
  5   4   4    3
(End)

			a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A100881, A100882, A100884.

Cf. A000085, A000219, A003293, A006951, A100471, A323582.

b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
       add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)

More terms from Vladeta Jovovic, Nov 23 2004

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518

Offset: 0

N. J. A. Sloane

An ordinary partition is a row of numbers in nondecreasing order whose sum is n. Here the numbers are in a three-dimensional pile, nondecreasing in the x-, y- and z-directions.
Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.
Equals A000041 convolved with A002836: [1, 0, 2, 5, 12, 24, 56, 113, ...] and row sums of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

			Examples for n=2 and n=3.
a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.
a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.
From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(4) = 26 solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((21))           ((22))
         ((1)(1))    ((111))          ((31))
         ((1))((1))  ((2)(1))         ((211))
                     ((11)(1))        ((1111))
                     ((2))((1))       ((2)(2))
                     ((1)(1)(1))      ((3)(1))
                     ((11))((1))      ((21)(1))
                     ((1)(1))((1))    ((11)(11))
                     ((1))((1))((1))  ((111)(1))
                                      ((2))((2))
                                      ((3))((1))
                                      ((2)(1)(1))
                                      ((21))((1))
                                      ((11))((11))
                                      ((11)(1)(1))
                                      ((111))((1))
                                      ((2)(1))((1))
                                      ((1)(1)(1)(1))
                                      ((11)(1))((1))
                                      ((2))((1))((1))
                                      ((1)(1))((1)(1))
                                      ((1)(1)(1))((1))
                                      ((11))((1))((1))
                                      ((1)(1))((1))((1))
                                      ((1))((1))((1))((1))
(End)

P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Suresh Govindarajan, Table of n, a(n) for n = 0..72
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666.
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, arXiv:1406.5605 [cond-mat.stat-mech], 2014; J. Stat. Phys. 158 (2015) 950-967.
Suresh Govindarajan, Solid Partitions Project Dec 14, 2010.
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
P. A. MacMahon, Combinatory analysis.
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003; J. Phys. A 36 (2003), no. 24, 6651-6659.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
Eric Weisstein's World of Mathematics, Solid Partition
Wikipedia, Solid partition
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A007326, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim).
Cf. A161564. - Gary W. Adamson, Jun 13 2009
Cf. A001970, A002974, A003293, A007713, A114736, A117433, A323657.

planePtns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And@@(GreaterEqual@@@Transpose[PadRight[#]])&],{ptn,IntegerPartitions[n]}];
solidPtns[n_]:=Join@@Table[Select[Tuples[planePtns/@y],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])&],{y,IntegerPartitions[n]}];
Table[Length[solidPtns[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from the Mustonen and Rajesh article, May 02 2003
a(51)-a(62) found by Suresh Govindarajan and students, Dec 14 2010
a(63)-a(68) found by Suresh Govindarajan and students, Jun 01 2011
a(69)-a(72) found by Suresh Govindarajan and Srivatsan Balakrishnan, Jan 03 2013

A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 17, 28, 41, 75, 122, 192, 314, 484, 771, 1216, 1861, 2848, 4395, 6610, 10037

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
From Gus Wiseman, Jan 17 2019: (Start)
Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:
  6   51   42   321
.
  5   4   41   31   32   31   22   221   211
  1   2   1    2    1    11   2    1     11
.
  3   21   21   111
  2   2    11   11
  1   1    1    1
(End)

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{2},{1,2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{1,1},{1,1,1}}
   {{1,2},{1,2,2}}
   {{2,2},{1,2,2}}

Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646.

Cf. A000085, A138178, A323436.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,UnsameQ@@Transpose[PadRight[#]],And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}] (* Gus Wiseman, Jan 18 2019 *)

a(11)-a(17) from Gus Wiseman, Jan 18 2019

a(18)-a(21) from Robert Price, Jun 21 2021

A323429 Number of rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 58, 81, 124, 169, 257, 345, 501, 684, 968, 1304, 1830, 2452, 3387, 4541, 6188, 8257, 11193, 14865, 19968, 26481, 35341, 46674, 62007, 81611, 107860, 141602, 186292, 243800, 319610, 416984, 544601, 708690, 922472, 1197018, 1553442

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing.

			The a(5) = 14 matrices:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323430, A323431, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 15, 22, 37, 56, 92, 133, 210, 310, 466, 696, 1013, 1495, 2160, 3141, 4495, 6462, 9172, 13024, 18387, 25840, 36213, 50500, 70280, 97302, 134522, 185105, 254245, 347938, 475036, 646676, 878145, 1189468, 1607095, 2166672, 2913794, 3910741

Offset: 0

Vaclav Kotesovec, Oct 03 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A003293, A035528, A161870, A262736, A292038.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d::even, 0, d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(-1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2) * A * Zeta(3)^(5/36) / (2^(2/3) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 09 2017

A323436 Number of plane partitions whose parts are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 7, 2, 4, 2, 3, 1, 7, 2, 5, 2, 2, 1, 8, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 4, 3, 2, 4, 1, 7, 5, 2, 1, 8, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly decreasing.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(120) = 12 plane partitions:
  32111
.
  311   321   3111   3211
  21    11    2      1
.
  31   32   311   321
  21   11   2     1
  1    1    1     1
.
  31   32
  2    1
  1    1
  1    1
.
  3
  2
  1
  1
  1

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A296188, A299968.

Cf. A323300, A323429, A323437, A323438, A323439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,100}]

A089299 Number of square plane partitions of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 25 2003

nonn

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.

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A008763, A001970, A089292.

Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 16 2019 *)

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

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

A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 4, 1, 4, 4, 2, 1, 12, 2, 3, 2, 4, 1, 5, 2, 7, 2, 2, 1, 10, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 5, 2, 1, 10, 2

Offset: 1

Gus Wiseman, Jan 16 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(96) = 19 tableaux:
  111112
.
  111   1111   1112   11111   11112
  112   12     11     2       1
.
  11   111   111   112   1111   1112
  11   11    12    11    1      1
  12   2     1     1     2      1
.
  11   11   111   112
  11   12   1     1
  1    1    1     1
  2    1    2     1
.
  11   12
  1    1
  1    1
  1    1
  2    1
.
  1
  1
  1
  1
  1
  2

Wikipedia, Young tableau

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A299968, A323300, A323430, A323436, A323438, A323450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@LessEqual@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Sum_{A056239(n) = k} a(k) = A323450(n).

A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 12, 16, 22, 27, 36, 44, 57, 72, 89, 110, 139, 170, 210, 261, 318, 390, 478, 581, 705, 860, 1036, 1252, 1511, 1816, 2178, 2618, 3127, 3743, 4471, 5330, 6347, 7564, 8984, 10674, 12669, 15016, 17780, 21050, 24868, 29371, 34655, 40836, 48080

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are strictly decreasing.

			The a(8) = 12 matrices:
  [8] [7 1] [6 2] [5 3] [5 2 1] [4 3 1]
.
  [7] [6] [5] [3 2]
  [1] [2] [3] [2 1]
.
  [5] [4]
  [2] [3]
  [1] [1]
The a(10) = 22 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [5 2] [4 2] [4 3]
  [1] [2] [3] [4] [2 1] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323431, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[Select[IntegerPartitions[#,{k}],UnsameQ@@#&]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323432 Number of semistandard rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 20, 30, 42, 59, 79, 112, 146, 199, 264, 350, 455, 603, 774, 1010, 1297, 1668, 2124, 2724, 3441, 4372, 5513, 6955, 8718, 10960, 13670, 17091, 21264, 26454, 32786, 40667, 50215, 62048, 76435, 94126

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows are weakly decreasing and the columns are strictly decreasing.

			The a(6) = 15 matrices:
  [6] [51] [42] [411] [33] [321] [3111] [222] [2211] [21111] [111111]
.
  [5] [4] [22]
  [1] [2] [11]
.
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A089299, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323431, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

cp's OEIS Frontend

A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

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A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

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A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

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A323429 Number of rectangular plane partitions of n.

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A262811 Expansion of Product_{k>=1} 1/(1-x^(2*k-1))^(2*k-1).

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A323436 Number of plane partitions whose parts are the prime indices of n.

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A089299 Number of square plane partitions of n.

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A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

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A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).