A365662 -id:A365662 - 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

A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 2, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 2, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 6, 5, 6, 5, 6, 4, 5, 10, 12, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 12, 15, 7, 6, 8, 7, 8, 8, 8, 7, 8, 6, 7, 15

Offset: 0

Peter Bala, Mar 30 2017

Compare with A284592.

			Square array begins
  n\k| 0  1  2  3  4  5  6   7   8   9  10  11  12  13
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0  | 1  1  1  2  2  3  4   5   6   8  10  12  15  18: A000009
  1  | 1  0  1  1  1  2  2   3   3   5   5   7   8  10: A096765
  2  | 1  1  0  1  2  2  2   3   4   5   6   7   9  11: A015744
  3  | 2  1  1  2  2  3  4   6   6   8   9  12  15  18
  4  | 2  1  2  2  2  3  5   5   7   9  10  14  15  19
  5  | 3  2  2  3  3  6  6   8   9  12  16  19  22  28
  6  | 4  2  2  4  5  6  8   9  11  16  18  22  27  33
  7  | 5  3  3  6  5  8  9  14  16  20  23  29  34  41
  ...
T(3,7) = 6: the six pairs of partitions of 3 and 7 into distinct parts and with no parts in common are (3, 7), (3, 6 + 1), (3, 5 + 2), (3, 4 + 2 + 1), (2 + 1, 7) and (2 + 1, 4 + 3).

Alois P. Heinz, Antidiagonals n = 0..200, flattened
H. S. Wilf, Lectures on Integer Partitions

Rows n=0..2 give A000009, A096765, A015744.
Main diagonal gives A365662.
Antidiagonal sums give A032302.
Cf. A284592, A322210.

# A284593 as a square array
ser := taylor(taylor(mul(1 + x^j + y^j, j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand((x^i+1)*b(n-i, min(n-i, i-1)))))
    end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Aug 24 2019

nmax = 12; M = CoefficientList[#, y][[;; nmax+1]]& /@ (Product[1 + x^j + y^j, {j, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& // Expand);
T[n_, k_] := M[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

O.g.f. Product_{j >= 1} (1 + x^j + y^j) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A032302.

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).

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.

A382980 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k - y^k).

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 6, 4, 10, 6, 14, 10, 14, 10, 20, 6, 22, 2, 10, 14, 16, -32, 14, 6, -26, -20, 12, -56, 28, -2, -38, 96, 56, -38, 200, 298, 82, 338, 460, 446, 666, 852, 456, 1580, 1172, 1048, 1608, 2426, 1236, 2810, 2222, 2824, 2066, 3716, 1612, 5498

Offset: 0

Seiichi Manyama, Apr 11 2025

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..130

Main diagonal of A382975.

Cf. A365662.

nmax := 60; prec := 2*nmax + 10; Rx := PowerSeriesRing(Rationals(), prec); Rxy := PowerSeriesRing(Rx, prec); P := Rxy!1; for k in [1..prec] do P *:= (1 + x^k - y^k); end for; seq := [Coefficient(Coefficient(P, n), n) : n in [0..nmax]]; print seq; // Vincenzo Librandi, Apr 13 2025

a[n_]:=SeriesCoefficient[Product[(1+x^k-y^k),{k,1,n}],{x,0,n},{y,0,n}];Table[a[n],{n,0,60}] (* Vincenzo Librandi, Apr 13 2025 *)

A382948 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k)^k.

Original entry on oeis.org

1, 0, 2, 18, 50, 190, 536, 1644, 4432, 12876, 33560, 89118, 227734, 572578, 1409602, 3424996, 8150818, 19152532, 44455758, 101565172, 229712612, 513207144, 1134650028, 2481664146, 5379539720, 11545719858, 24574548632, 51855844492, 108559596182

Offset: 0

Ilya Gutkovskiy, Apr 09 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A026007, A365662.

Table[SeriesCoefficient[Product[(1 + x^k + y^k)^k, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 28}]

A382950 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k)^n.

Original entry on oeis.org

1, 0, 6, 102, 1342, 20030, 306852, 4783534, 75873934, 1220259306, 19837742836, 325375411438, 5376744428812, 89412908941096, 1494992390431000, 25114561595879252, 423649216254936110, 7172523302899053230, 121828099966104173892, 2075321708914763792740

Offset: 0

Ilya Gutkovskiy, Apr 09 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A270913, A365662.

Table[SeriesCoefficient[Product[(1 + x^k + y^k)^n, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 19}]

a(n) ~ c * d^n / n, where d = 17.95370167248385600263... and c = 0.0600668009236121058... - Vaclav Kotesovec, Apr 10 2025

A382959 a(n) = (n!)^2 * [(x*y)^n] Product_{k>=1} (1 + (x^k + y^k)/k!).

Original entry on oeis.org

1, 0, 0, 6, 8, 130, 342, 2590, 21240, 167730, 1874930, 46128610, 417338462, 5163377570, 542567363366, 3984766703746, 42736508056760, 681324935577810, 127138303030260258, 1011227775808000450, 14280379156264610778, 276342548314653322270, 12566141342987866203746

Offset: 0

Ilya Gutkovskiy, Apr 10 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A007837, A108796, A365662.

Table[(n!)^2 SeriesCoefficient[Product[(1 + (x^k + y^k)/k!), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 22}]

cp's OEIS Frontend

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

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A284593 Square array read by antidiagonals: T(n,k) = the number of pairs of partitions of n and k respectively, such that each partition is composed of distinct parts and the pair of partitions have no part in common.

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

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

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A382980 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k - y^k).

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A382948 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k)^k.

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A382950 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k)^n.

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A382959 a(n) = (n!)^2 * [(x*y)^n] Product_{k>=1} (1 + (x^k + y^k)/k!).

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