A101509 -id:A101509 - cp's OEIS frontend

A323432 Number of semistandard rectangular plane partitions of n.

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}]

A323431 Number of strict rectangular plane partitions of n.

Original entry on oeis.org

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}]

A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 20, 24, 31, 38, 48, 59, 72, 86, 106, 125, 150, 180, 213, 250, 296, 347, 407, 477, 555, 645, 751, 869, 1003, 1161, 1334, 1534, 1763, 2018, 2306, 2637, 3002, 3418, 3886, 4409, 4994, 5659, 6390, 7214, 8135, 9160, 10300, 11580, 12990

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(10) = 20 split partitions:
  [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 3]
  [1] [2] [3] [4] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Cf. A000005, A000219, A101509, A117433, A316245, A319066, A319794, A323295, A323301, A323431, A323433.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, numtheory[tau](t), b(n, i-1, t)+
         b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
     If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
     b[n-i, Min[n-i, i-1], t+1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.

A101508 Product of binomial matrix and the Mobius matrix A051731.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 3, 1, 16, 8, 6, 4, 1, 32, 16, 11, 10, 5, 1, 64, 32, 21, 20, 15, 6, 1, 128, 64, 42, 36, 35, 21, 7, 1, 256, 128, 85, 64, 70, 56, 28, 8, 1, 512, 256, 171, 120, 127, 126, 84, 36, 9, 1, 1024, 512, 342, 240, 220, 252, 210, 120, 45, 10, 1, 2048, 1024, 683, 496, 385, 463, 462, 330, 165, 55, 11, 1

Offset: 0

Paul Barry, Dec 05 2004

Row sums are A101509. Diagonal sums are A101510.
The matrix inverse appears to be A128313. - R. J. Mathar, Mar 22 2013
Read as upper triangular matrix, this can be seen as "recurrences in A135356 applied to A023531" [Paul Curtz, Mar 03 2017]. - The columns are: A000079, A131577, A024495, A000749, A139761, ... Column n differs after the (n+1)-th nonzero term on from the binomial coefficients C(k,n). - M. F. Hasler, Mar 05 2017

			Rows begin
  1;
  2,1;
  4,2,1;
  8,4,3,1;
  16,8,6,4,1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

A101508 := proc(n,k)
    a := 0 ;
    for i from 0 to n do
        if modp(i+1,k+1) = 0 then
            a := a+binomial(n,i) ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Mar 22 2013

t[n_, k_] := Sum[If[Mod[i + 1, k + 1] == 0, Binomial[n, i], 0], {i, 0, n}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

T(n,k)=sum(i=0,n, if((i+1)%(k+1)==0, binomial(n, i))) \\ M. F. Hasler, Mar 05 2017

T(n, k) = Sum_{i=0..n} if(mod(i+1, k+1)=0, binomial(n, i), 0).

Rows have g.f. x^k/((1-x)^(k+1)-x^(k+1)).

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}]

A323868 Number of matrices of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 6, 26, 225, 1082, 18732, 94586, 2183340, 21261783, 408990252, 3245265146, 168549405570, 1053716696762, 42565371881772, 921132763911412, 26578273409906775, 260741534058271802, 20313207979541071938, 185603174638656822266, 16066126777466305218690

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

			The 42 matrices of size 4 whose entries cover {1,2}:
  1222 2111 1122 2211 1212 2121 1221 2112 1112 2221 1121 2212 1211 2122
.
  12  21  11  22  12  21  12  21  11  22  11  22  12  21
  22  11  22  11  12  21  21  12  12  21  21  12  11  22
.
  1   2   1   2   1   2   1   2   1   2   1   2   1   2
  2   1   1   2   2   1   2   1   1   2   1   2   2   1
  2   1   2   1   1   2   2   1   1   2   2   1   1   2
  2   1   2   1   2   1   1   2   2   1   1   2   1   2
The 18 matrices of size 4 whose entries cover {1,2} with multiplicities {2,2}:
  [1 1 2 2] [2 2 1 1] [1 2 1 2] [2 1 2 1] [1 2 2 1] [2 1 1 2]
.
  [1 1] [2 2] [1 2] [2 1] [1 2] [2 1]
  [2 2] [1 1] [1 2] [2 1] [2 1] [1 2]
.
  [1] [2] [1] [2] [1] [2]
  [1] [2] [2] [1] [2] [1]
  [2] [1] [1] [2] [2] [1]
  [2] [1] [2] [1] [1] [2]

Alois P. Heinz, Table of n, a(n) for n = 1..424

Cf. A000005, A000670, A060223, A101509.

Cf. A323351, A323869, A323870, A323871.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> b(n)*numtheory[tau](n):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 04 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
Table[Length[nrmmats[n]],{n,6}]
Table[DivisorSigma[0, n]*Sum[k! StirlingS2[n, k], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 05 2019 *)

a(n) = numdiv(n)*sum(k=0, n, stirling(n, k, 2)*k!); \\ Michel Marcus, Feb 05 2019

a(n) = A000005(n) * A000670(n).

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).

A324914 a(n) = Sum_{k=1..n} 2^k * tau(k), where tau(k) = A000005(k).

Original entry on oeis.org

2, 10, 26, 74, 138, 394, 650, 1674, 3210, 7306, 11402, 35978, 52362, 117898, 248970, 576650, 838794, 2411658, 3460234, 9751690, 18140298, 34917514, 51694730, 185912458, 286575754, 555011210, 1091882122, 2702494858, 3776236682, 12366171274

Offset: 1

Vaclav Kotesovec, Mar 18 2019

nonn

Partial sums of A323351 with n=0 term of A323351 omitted. - Robert Israel, Jun 27 2019

Robert Israel, Table of n, a(n) for n = 1..3314
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A000005, A006218, A101509, A160399, A324915, A323351.

ListTools:-PartialSums([seq(2^k*numtheory:-tau(k),k=1..100)]); # Robert Israel, Jun 27 2019

Accumulate[Table[2^k*DivisorSigma[0, k], {k, 1, 30}]]

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).

cp's OEIS Frontend

A323432 Number of semistandard rectangular plane partitions of n.

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A323431 Number of strict rectangular plane partitions of n.

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A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

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A101508 Product of binomial matrix and the Mobius matrix A051731.

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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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A323868 Number of matrices of size n whose entries cover an initial interval of positive integers.

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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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