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

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

Author

Gus Wiseman, Oct 08 2023

Keywords

• nonn

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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

Formula

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