A323429 -id:A323429 - cp's OEIS frontend

A323431 Number of strict rectangular plane partitions of n.

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 21, 25, 33, 41, 53, 65, 81, 97, 121, 143, 173, 215, 255, 305, 367, 441, 527, 637, 751, 899, 1067, 1269, 1491, 1775, 2071, 2439, 2875, 3357, 3911, 4577, 5309, 6177, 7171, 8305, 9609, 11151

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

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

			The a(10) = 21 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] [4 2] [4 3]
  [1] [2] [3] [4] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

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

Cf. A323301, A323307, A323429, A323430, A323432, A323435, A323436, A323438.

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

A323529 Number of strict square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 55, 69, 79, 95, 109, 129, 145, 169, 189, 217, 241, 273, 301, 339, 371, 413, 451, 499, 541, 595, 643, 703, 757, 823, 925, 999, 1107, 1229, 1387, 1559, 1807, 2071, 2453, 2893, 3451, 4109, 5011

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(12) = 5 strict square plane partitions:
  [12]
.
  [1 2] [1 2] [1 3] [1 4]
  [3 6] [4 5] [2 6] [2 5]
The a(15) = 13 strict square plane partitions:
  [15]
.
  [7 5] [8 4] [9 3] [6 5] [7 4] [9 2] [6 4] [7 3] [8 2] [6 3] [6 3] [7 2]
  [2 1] [2 1] [2 1] [3 1] [3 1] [3 1] [3 2] [4 1] [4 1] [4 2] [5 1] [5 1]

Alois P. Heinz, Table of n, a(n) for n = 0..7500

Cf. A000219, A008289, A039622, A047966, A089299, A101509, A319066, A323429, A323434, A323522, A323530.

h:= proc(n) h(n):= (n^2)!*mul(k!/(n+k)!, k=0..n-1) end:
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, `if`(issqr(t), h(isqrt(t)), 0),
         b(n, i-1, t) +b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 24 2019

Table[Sum[Length[Select[Union[Sort/@Tuples[Reverse/@IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
h[n_] := (n^2)! Product[k!/(k+n)!, {k, 0, n-1}];
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0, If[n == 0, If[IntegerQ[ Sqrt[t]], h[Sqrt[t]], 0], b[n-i, Min[n-i, i-1], t+1] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 70] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(n) = Sum_{j>=0} A039622(j) * A008289(n,j^2). - Alois P. Heinz, Jan 24 2019

More terms from Alois P. Heinz, Jan 24 2019

A323435 Number of rectangular plane partitions of n with no repeated rows or columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 8, 13, 15, 28, 33, 52, 69, 101, 133, 202, 256, 369, 506, 688, 935, 1295, 1736, 2355, 3184, 4284, 5745, 7722, 10281, 13691, 18316, 24168, 32058, 42389, 55915, 73542, 96753, 126709, 166079, 217017, 283258

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 and with no repeated rows or columns.

			The a(7) = 13 plane partitions:
  [7] [4 3] [5 2] [6 1] [4 2 1]
.
  [6] [5] [3 2] [4 1] [4] [2 2] [3 1]
  [1] [2] [1 1] [1 1] [3] [2 1] [2 1]
.
  [4]
  [2]
  [1]

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

Cf. A323301, A323307, A323429, A323430, A323431, A323432, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And[UnsameQ@@#,UnsameQ@@Transpose[#],And@@(OrderedQ[#,GreaterEqual]&/@Transpose[#])]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,20}]

A306318 Number of square twice-partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 10, 12, 19, 24, 39, 49, 73, 104, 151, 212, 317, 443, 638, 936, 1296, 1841, 2635, 3641, 5069, 7176, 9884, 13614, 19113, 26162, 36603, 50405, 70153, 96176, 135388, 184753, 257882, 353587, 494653, 671992, 934905, 1272195, 1762979, 2389255

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is square if the number of parts is equal to the number of parts in each part.

			The a(10) = 19 square twice-partitions:
  ((ten))  ((32)(32))  ((211)(111)(111))
           ((32)(41))
           ((33)(22))
           ((33)(31))
           ((41)(32))
           ((41)(41))
           ((42)(22))
           ((42)(31))
           ((43)(21))
           ((44)(11))
           ((51)(22))
           ((51)(31))
           ((52)(21))
           ((53)(11))
           ((61)(21))
           ((62)(11))
           ((71)(11))

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

Cf. A000219, A001970, A063834 (twice-partitions), A089299 (square plane partitions), A279787, A305551, A306017, A306319 (rectangular twice-partitions), A319066, A323429, A323531 (square partitions of partitions).

Table[Sum[Length[Union@@(Tuples[IntegerPartitions[#,{k}]&/@#]&/@IntegerPartitions[n,{k}])],{k,0,Sqrt[n]}],{n,0,20}]

A323530 Number of square plane partitions of n with strictly decreasing rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 25, 30, 39, 46, 58, 67, 82, 94, 112, 127, 149, 168, 194, 218, 251, 282, 324, 368, 425, 489, 573, 670, 797, 952, 1148, 1392, 1703, 2086, 2568, 3168, 3908, 4823, 5947, 7318, 8986, 11012, 13443, 16371, 19866

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(12) = 8 plane partitions:
  [12]
.
  [5 4] [6 3] [7 2] [5 3] [6 2] [4 3] [5 2]
  [2 1] [2 1] [2 1] [3 1] [3 1] [3 2] [4 1]

Cf. A000219, A089299, A306320, A323429, A323434, A323439, A323522, A323529, A323530, A323531.

Table[Sum[Length[Select[Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn],And@@Greater@@@#&&And@@Greater@@@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}]

A382923 Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 5, 7, 0, 4, 13, 16, 16, 0, 5, 14, 33, 40, 35, 0, 6, 29, 70, 105, 100, 75, 0, 7, 27, 88, 207, 292, 244, 159, 0, 8, 51, 152, 336, 604, 758, 576, 334, 0, 9, 44, 206, 588, 1161, 1749, 1920, 1329, 696, 0, 10, 79, 300, 882, 2076, 3685, 4924, 4802, 3028, 1442

Offset: 0

John Tyler Rascoe, Apr 09 2025

For some m > 0, an m-composition of n is a rectangular array of nonnegative integers with m rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.

			Square array begins:
   1,   0,   0,   0,    0,    0, ...
   1,   2,   3,   4,    5,    6, ...
   3,   5,  13,  14,   29,   27, ...
   7,  16,  33,  70,   88,  152, ...
  16,  40, 105, 207,  336,  588, ...
  35, 100, 292, 604, 1161, 2076, ...
  ...
A(2,0) = 3 counts:
  [2],  [1,1],  [1]
                [1].
A(2,1) = 5 counts:
  [2]   [0]   [1]   [1]   [0]
  [0],  [2],  [1]   [0]   [1]
              [0],  [1],  [1].

John Tyler Rascoe, Antidiagonals n = 0..30, flattened

Cf. A038207, A101509 (column k=0), A181331, A261780, A323429, A382924 (main diagonal).

G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
G_tx(10)

G.f.: G(t,x) = 1 + Sum_{m>0} -1 + 1/(1 + t^m - (t + x/(1 - x))^m).

A382924 Number of m-compositions of n with n zeros.

Original entry on oeis.org

1, 2, 13, 70, 336, 2076, 11091, 65210, 365661, 2159354, 11713047, 71427504, 392916687, 2245186352, 13527678851, 73679458270, 429472428457, 2553994191220, 14264421153074, 80483620074092, 489077890675807, 2768919905996888, 15394229582049408, 91794448088043258

Offset: 0

John Tyler Rascoe, Apr 09 2025

nonn

For some m > 0, an m-composition of n is a rectangular array of nonnegative integers with m rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.

			a(2) = 13 counts:
  [2]  [0]  [0]  [1]  [1]  [1]  [0]  [0]  [0]  [1][1]  [1][0]  [0][0]  [0][1]
  [0]  [2]  [0]  [1]  [0]  [0]  [1]  [1]  [0]  [0][0], [0][1], [1][1], [1][0].
  [0], [0], [2], [0]  [1]  [0]  [1]  [0]  [1]
                 [0], [0], [1], [0], [1], [1],

Cf. A038207, A101509, A181331, A261780, A323429, A382820, (main diagonal of A382923).

G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
A382924(max_n) ={my(A=G_tx(max_n)); vector(max_n,i,A[i,i])}
A382924(20)

a(n) = [(x*t)^n] 1 + Sum_{m>0} -1 + 1/(1 + t^m - (t + x/(1 - x))^m).

A306320 Number of square plane partitions of n with distinct row sums and distinct column sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 5, 5, 10, 11, 18, 21, 31, 37, 56, 70, 97, 134, 180, 247, 343, 462, 623, 850, 1128, 1509, 2004, 2649, 3467, 4590, 5958, 7814, 10161, 13287, 17208, 22495, 29129, 37997, 49229, 64098, 82940, 107868, 139390, 180737, 233214, 301527, 388018, 500058

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

			The a(12) = 21 square plane partitions with distinct row sums and distinct column sums:
[twelve]
.
[64][73][82][91][54][63][72][81][44][53][53][62][62][71][43][43][52][52][61]
[11][11][11][11][21][21][21][21][31][22][31][22][31][31][32][41][32][41][41]
.
[221]
[211]
[111]

Cf. A000219, A089299 (square plane partitions), A101509, A271619, A279785, A306318, A323429, A323529, A323530, A323531.

Table[Sum[Length[Select[Union[Reverse/@Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Total/@#&&UnsameQ@@Total/@If[#=={},{},Transpose[#]]&&And@@OrderedQ/@Reverse/@If[#=={},{},Transpose[#]]&]],{ptn,IntegerPartitions[n]}],{n,0,20}]

A323586 Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).

Original entry on oeis.org

1, 1, 2, 5, 8, 16, 30, 53, 89, 158, 265, 443, 735, 1197

Offset: 0

Gus Wiseman, Jan 20 2019

			The a(4) = 8 plane partitions with no repeated rows:
  4   31   22   211   1111
.
  3   21   111
  1   1    1
The a(6) = 30 plane partitions with no repeated columns:
  6   51   42   321
.
  5   4   41   3   31   32   31   22   21   221   211
  1   2   1    3   2    1    11   2    21   1     11
.
  4   3   31   2   21   22   21   111
  1   2   1    2   2    1    11   11
  1   1   1    2   1    1    1    1
.
  3   2   21   11
  1   2   1    11
  1   1   1    1
  1   1   1    1
.
  2   11
  1   1
  1   1
  1   1
  1   1
.
  1
  1
  1
  1
  1
  1

Cf. A000219, A003293 (strict rows), A114736 (strict rows and columns), A117433 (distinct entries), A299968, A319646 (no repeated rows or columns), A323429, A323436 (plane partitions of type), A323580, A323587.

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@@#,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}]

cp's OEIS Frontend

A323431 Number of strict rectangular plane partitions of n.

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A323529 Number of strict square plane partitions of n.

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A323435 Number of rectangular plane partitions of n with no repeated rows or columns.

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A306318 Number of square twice-partitions of n.

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A323530 Number of square plane partitions of n with strictly decreasing rows and columns.

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A382923 Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.

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A382924 Number of m-compositions of n with n zeros.

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PARI

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A306320 Number of square plane partitions of n with distinct row sums and distinct column sums.

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Mathematica

A323586 Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).

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