A086737 -id:A086737 - cp's OEIS frontend

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A355389 Number of unordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)

The version for compositions is A006516.
Without distinctness we get A086737.
The unordered version is A355390, without distinctness A001255.
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, A319794, A319910, A323433, A339006, A355385.

a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024

Table[Binomial[PartitionsP[n],2],{n,0,6}]

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

a(n) = binomial(A000041(n), 2) = A355390(n)/2.

A366317 Number of unordered pairs of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 15, 21, 36, 55, 78, 120, 171, 253, 378, 528, 741, 1081, 1485, 2080, 2926, 4005, 5460, 7503, 10153, 13695, 18528, 24753, 32896, 43956, 57970, 76245, 100576, 131328, 171405, 223446, 289180, 373680, 482653, 619941, 794430, 1017451, 1296855

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(1) = 1 through a(7) = 15 unordered pairs of strict partitions:
  {1,1}  {2,2}  {3,3}    {4,4}    {5,5}    {6,6}      {7,7}
                {3,21}   {4,31}   {5,32}   {6,42}     {7,43}
                {21,21}  {31,31}  {5,41}   {6,51}     {7,52}
                                  {32,32}  {42,42}    {7,61}
                                  {32,41}  {42,51}    {43,43}
                                  {41,41}  {51,51}    {43,52}
                                           {6,321}    {43,61}
                                           {42,321}   {52,52}
                                           {51,321}   {52,61}
                                           {321,321}  {61,61}
                                                      {7,421}
                                                      {43,421}
                                                      {52,421}
                                                      {61,421}
                                                      {421,421}

For non-strict partitions we have A086737.
The disjoint case is A108796, non-strict A260669.
The ordered version is A304990, disjoint A032302.
The ordered disjoint case is A365662.
Excluding constant pairs gives A366132.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000712, A007582, A054440, A064914, A260664.

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

a(n) = A000217(A000009(n)).

Composition of A000009 and A000217.

A370207 Number T(n,k) of unordered 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, 1, 1, 1, 2, 2, 1, 1, 6, 4, 3, 1, 1, 8, 10, 5, 3, 1, 1, 24, 18, 13, 6, 3, 1, 1, 30, 42, 23, 14, 6, 3, 1, 1, 74, 72, 55, 26, 15, 6, 3, 1, 1, 110, 146, 95, 61, 27, 15, 6, 3, 1, 1, 219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1, 309, 475, 351, 227, 115, 65, 28, 15, 6, 3, 1, 1

Offset: 0

Alois P. Heinz, Feb 11 2024

			T(4,0) = 6: (1111,22), (1111,4), (211,4), (22,31), (22,4), (31,4).
T(4,1) = 4: (1111,31), (211,22), (211,31), (4,4).
T(4,2) = 3: (1111,211), (22,22), (31,31).
T(4,3) = 1: (211,211).
T(4,4) = 1: (1111,1111).
Triangle T(n,k) begins:
    1;
    0,   1;
    1,   1,   1;
    2,   2,   1,   1;
    6,   4,   3,   1,  1;
    8,  10,   5,   3,  1,  1;
   24,  18,  13,   6,  3,  1,  1;
   30,  42,  23,  14,  6,  3,  1, 1;
   74,  72,  55,  26, 15,  6,  3, 1, 1;
  110, 146,  95,  61, 27, 15,  6, 3, 1, 1;
  219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1;
  ...

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

Column k=0 gives A260669.
Row sums and T(2n,n) give A086737.
Cf. A000041, A000217, A008284, A072233, A370005.

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:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(g(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= (n, k)-> (coeff(b(n$3), x, k)+coeff(g(n$2), x, k))/2:
seq(seq(T(n, k), k=0..n), n=0..12);

T(n,k) = (A370005(n,k) + A072233(n,k))/2.

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).

cp's OEIS Frontend

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

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A355389 Number of unordered pairs of distinct integer partitions of n.

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A366317 Number of unordered pairs of strict integer partitions of n.

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A370207 Number T(n,k) of unordered 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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A355390 Number of ordered pairs of distinct integer partitions of n.

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