A054440 -id:A054440 - cp's OEIS frontend

A274521 Number of odd partitions in the multiset of intersections of the set of partitions of n with itself; also number of distinct partitions in that multiset.

1, 1, 4, 8, 23, 44, 107, 190, 406, 722, 1394, 2383, 4434, 7342, 12901, 21162, 35754, 57286, 94294, 147980, 237716, 368255, 577038, 880400, 1358074, 2043017, 3097194, 4607048, 6882358, 10121400, 14937754, 21726770, 31695300, 45685964, 65909693, 94165650

Offset: 1

George Beck, Jun 26 2016

nonn

Let a(n) be the number of odd partitions in the multiset intersections of the set of partitions of n with itself.
Form the p(n) X p(n) matrix M of partitions of numbers ranging from 1 to n by taking the multiset intersections of all the pairs of partitions of n. Then, ignoring the empty set, the number of odd partitions in M equals the number of distinct partitions in M. (Proved in Wilf et al., "A pentagonal number sieve".)
By numerical experimentation, it seems a(n) is the convolution of A000009 (with offset 1) and A054440. (conjectured)

			For n=3, the partitions are 3, 21, 111. The multiset intersections are M = {{3, x, x}, {x, 21, 1}, {x, 1, 111}} (where x is the empty set), which fall into classes {{OD, y, y}, {y, D, OD}, {y, OD, O}}, where O means odd, D means distinct, OD means both, and y means neither. Thus a(3) = 4, the number of Os, which equals the number of Ds.

George Beck, Mathematica notebook
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, and Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Cf. A000009, A054440.

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

Original entry on oeis.org

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).

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.

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);

A304987 G.f.: Sum_{k>=0} A000041(k) * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Original entry on oeis.org

1, 0, 1, -2, 2, -6, 3, -16, 17, -34, 47, -78, 153, -178, 373, -530, 954, -1410, 2280, -3896, 5908, -9988, 15170, -25908, 40659, -65136, 105967, -169056, 276483, -435624, 712052, -1139814, 1839535, -2955466, 4745201, -7689672, 12303439, -19866340, 31904000

Offset: 0

Vaclav Kotesovec, May 23 2018

sign

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

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]*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 = 0.4455996319406557616008349987776746416976798533740571426884585957313974660...

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...

cp's OEIS Frontend

A274521 Number of odd partitions in the multiset of intersections of the set of partitions of n with itself; also number of distinct partitions in that multiset.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A304987 G.f.: Sum_{k>=0} A000041(k) * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula