A001255 -id:A001255 - cp's OEIS frontend

A304988 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k) * x^k.

1, 0, 3, 4, 16, 20, 67, 84, 231, 324, 735, 1026, 2265, 3086, 6199, 8880, 16564, 23390, 42378, 59496, 103588, 146376, 244278, 344186, 564013, 788168, 1255201, 1758400, 2738833, 3812242, 5846114, 8092092, 12200957, 16848156, 24991705, 34365176, 50392543

Offset: 0

Vaclav Kotesovec, May 23 2018

nonn

Cf. A000009, A000041, A001255, A054440, A304877, A305350.

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ 7^(3/2) * exp(Pi*sqrt(7*n/3)) / (768*n^2).

A366132 Number of unordered pairs of distinct strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 10, 15, 28, 45, 66, 105, 153, 231, 351, 496, 703, 1035, 1431, 2016, 2850, 3916, 5356, 7381, 10011, 13530, 18336, 24531, 32640, 43660, 57630, 75855, 100128, 130816, 170820, 222778, 288420, 372816, 481671, 618828, 793170, 1016025, 1295245

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(3) = 1 through a(8) = 15 pairs of strict partitions:
  {3,21}  {4,31}  {5,32}   {6,42}    {7,43}    {8,53}
                  {5,41}   {6,51}    {7,52}    {8,62}
                  {41,32}  {51,42}   {7,61}    {8,71}
                           {6,321}   {52,43}   {62,53}
                           {42,321}  {61,43}   {71,53}
                           {51,321}  {61,52}   {71,62}
                                     {7,421}   {8,431}
                                     {43,421}  {8,521}
                                     {52,421}  {53,431}
                                     {61,421}  {53,521}
                                               {62,431}
                                               {62,521}
                                               {71,431}
                                               {71,521}
                                               {521,431}

For subsets instead of partitions we have A006516, non-disjoint A003462.
The disjoint case is A108796, non-strict A260669.
For non-strict partitions we have A355389.
The ordered disjoint case is A365662, non-strict A054440.
The ordered version is 2*a(n).
Including equal pairs or twins gives A366317, ordered A304990.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A161680 and A000217 count 2-subsets of {1..n}.
Cf. A000124, A001255, A006827, A032302, A064914, A365661, A365663.

Table[Length[Subsets[Select[IntegerPartitions[n],UnsameQ@@#&],{2}]],{n,0,30}]

a(n) = binomial(A000009(n),2).

A368564 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) = o(p,q), where d and o are distance functions; see Comments.

Original entry on oeis.org

1, 2, 3, 7, 15, 43, 57, 60, 82, 134, 184, 247, 331, 451, 562, 771, 985, 1277, 1630, 2071, 2640, 3344, 4119, 5195, 6514, 8062

Offset: 1

Clark Kimberling, Dec 31 2023

The definition of d depends on the greedy ordering of the partitions p(i) of n; that is, p(1) >= p(2) >= ... >= p(k), where k = A000041(n); see A366156. The ordinal distance o is defined by o(p(i),p(j)) = |i-j|.

			The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 7 pairs (p,q) for which d(p,q) = o(p,q), so a(4) = 7.

Cf. A000041, A001255, A366156, A368565, A368566.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

a(n) + A368565(n) + A368566(n) = A001255(n) for n >= 1.

A370005 Number T(n,k) of ordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 4, 3, 1, 1, 12, 7, 4, 1, 1, 16, 19, 8, 4, 1, 1, 48, 35, 23, 9, 4, 1, 1, 60, 83, 43, 24, 9, 4, 1, 1, 148, 143, 106, 47, 25, 9, 4, 1, 1, 220, 291, 186, 115, 48, 25, 9, 4, 1, 1, 438, 511, 397, 210, 119, 49, 25, 9, 4, 1, 1, 618, 949, 697, 444, 219, 120, 49, 25, 9, 4, 1, 1

Offset: 0

Alois P. Heinz, Feb 07 2024

			T(4,0) = 12: (1111,22), (1111,4), (211,4), (22,1111), (22,31), (22,4), (31,22), (31,4), (4,1111), (4,211), (4,22), (4,31).
T(4,1) = 7: (1111,31), (211,22), (211,31), (22,211), (31,1111), (31,211), (4,4).
T(4,2) = 4: (1111,211), (211,1111), (22,22), (31,31).
T(4,3) = 1: (211,211).
T(4,4) = 1: (1111,1111).
Triangle T(n,k) begins:
    1;
    0,   1;
    2,   1,   1;
    4,   3,   1,   1;
   12,   7,   4,   1,   1;
   16,  19,   8,   4,   1,  1;
   48,  35,  23,   9,   4,  1,  1;
   60,  83,  43,  24,   9,  4,  1, 1;
  148, 143, 106,  47,  25,  9,  4, 1, 1;
  220, 291, 186, 115,  48, 25,  9, 4, 1, 1;
  438, 511, 397, 210, 119, 49, 25, 9, 4, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives A054440.
Row sums and T(2n,n) give A001255.
Cf. A000041, A000290, A260669, A370207.

b:= proc(n, m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
      add(add(expand(b(sort([n-i*j, m-i*h])[], i-1)*
       x^min(j, h)), h=0..m/i), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$3)):
seq(T(n), n=0..12);

A141210 Triangle read by rows, A140207^2.

Original entry on oeis.org

1, 2, 1, 4, 3, 4, 7, 6, 10, 9, 12, 11, 20, 24, 25, 19, 18, 34, 45, 60, 49, 30, 29, 56, 78, 115, 126, 121, 45, 44, 86, 123, 190, 231, 286, 225, 67, 66, 130, 189, 300, 385, 528, 555, 484, 97, 96, 190, 279, 450, 595, 858, 1005, 1144, 900

Offset: 0

Gary W. Adamson & Roger L. Bagula, Jun 13 2008

Left border of the triangle = A000070; right border = A001255.

			First few rows of the triangle are:
1;
2, 1;
4, 3, 4;
7, 6, 10, 9;
12, 11, 20, 24, 25;
19, 18, 34, 45, 60, 49;
30, 29, 56, 78, 115, 126, 121;
...

Cf. A001255, A141211 (row sums), A140207, A000070.

A140207^2 as an infinite lower triangular matrix.

A355390 Number of ordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 2, 6, 20, 42, 110, 210, 462, 870, 1722, 3080, 5852, 10100, 18090, 30800, 53130, 87912, 147840, 239610, 392502, 626472, 1003002, 1573770, 2479050, 3831806, 5931660, 9057090, 13819806, 20834660, 31399212, 46806122, 69697452, 102870306, 151523790, 221488806

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(3) = 6 pairs:
  .  .  (11)(2)  (21)(3)
        (2)(11)  (3)(21)
                 (111)(3)
                 (3)(111)
                 (111)(21)
                 (21)(111)

Without distinctness we have A001255, unordered A086737.
The version for compositions is A020522, unordered A006516.
The unordered version is A355389.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319910, A323433, A323583, A339006, A355385.

Table[Length[Select[Tuples[IntegerPartitions[n],2],UnsameQ@@#&]],{n,0,15}]

a(n) = 2*binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

a(n) = 2*A355389(n) = 2*binomial(A000041(n), 2).

A304989 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Original entry on oeis.org

1, 0, 3, 2, 16, 10, 59, 32, 187, 90, 519, 152, 1439, 164, 3525, -246, 8904, -2500, 21748, -10836, 53918, -36508, 131424, -115266, 328703, -336608, 812615, -957464, 2046225, -2634166, 5152190, -7145682, 13121677, -19039178, 33473773, -50395004, 86035125

Offset: 0

Vaclav Kotesovec, May 23 2018

sign

Cf. A000009, A000041, A001255, A054440, A304877, A304990.

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]^2*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ (-1)^n * c * d^n, where d = 1.6096199212376592810929072080593393678131347423108390218672748044914523428584..., c = 3.049014588253509415528984781833089943634060493523166258285691300445092167...

A383354 Squares of plane partition numbers.

Original entry on oeis.org

1, 1, 9, 36, 169, 576, 2304, 7396, 25600, 79524, 250000, 737881, 2187441, 6175225, 17363889, 47320641, 127622209, 336135556, 876219201, 2240128900, 5666777284, 14112014436, 34772925625, 84554753089, 203576025636, 484461937089, 1142215875025, 2665572144964, 6166451098756

Offset: 0

Ilya Gutkovskiy, Apr 24 2025

nonn

Eric Weisstein's World of Mathematics, Plane Partition

Cf. A000219, A001255, A304990.

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]^2

a(n) = [(x*y)^n] Product_{k>=1} 1 / ((1 - x^k) * (1 - y^k))^k.

a(n) = A000219(n)^2.

cp's OEIS Frontend

A304988 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k) * x^k.

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A366132 Number of unordered pairs of distinct strict integer partitions of n.

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A368564 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) = o(p,q), where d and o are distance functions; see Comments.

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A370005 Number T(n,k) of ordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

A141210 Triangle read by rows, A140207^2.

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A355390 Number of ordered pairs of distinct integer partitions of n.

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A304989 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

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A383354 Squares of plane partition numbers.

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