A361801 Number of nonempty subsets of {1..n} with median n/2.

0, 0, 1, 1, 4, 4, 14, 14, 49, 49, 175, 175, 637, 637, 2353, 2353, 8788, 8788, 33098, 33098, 125476, 125476, 478192, 478192, 1830270, 1830270, 7030570, 7030570, 27088870, 27088870, 104647630, 104647630, 405187825, 405187825, 1571990935, 1571990935

Offset: 0

Author

Gus Wiseman, Apr 07 2023

Keywords

• nonn
• easy

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The subset {1,2,3,5} of {1..5} has median 5/2, so is counted under a(5).
The subset {2,3,5} of {1..6} has median 6/2, so is counted under a(6).
The a(0) = 0 through a(7) = 14 subsets:
  .  .  {1}  {1,2}  {2}      {1,4}      {3}          {1,6}
                    {1,3}    {2,3}      {1,5}        {2,5}
                    {1,2,3}  {1,2,3,4}  {2,4}        {3,4}
                    {1,2,4}  {1,2,3,5}  {1,3,4}      {1,2,5,6}
                                        {1,3,5}      {1,2,5,7}
                                        {1,3,6}      {1,3,4,5}
                                        {2,3,4}      {1,3,4,6}
                                        {2,3,5}      {1,3,4,7}
                                        {2,3,6}      {2,3,4,5}
                                        {1,2,4,5}    {2,3,4,6}
                                        {1,2,4,6}    {2,3,4,7}
                                        {1,2,3,4,5}  {1,2,3,4,5,6}
                                        {1,2,3,4,6}  {1,2,3,4,5,7}
                                        {1,2,3,5,6}  {1,2,3,4,6,7}
		

Crossrefs

A bisection is A079309.

The case with n's has bisection A057552.

The case without n's is A100066, bisection A006134.

A central diagonal of A231147.

A version for partitions is A361849.

For mean instead of median we have A362046.

A000975 counts subsets with integer median, for mean A327475.

A007318 counts subsets by length.

A013580 appears to count subsets by median, by mean A327481.

A360005(n)/2 represents the median statistic for partitions.

Cf. A024718, A325347, A359893, A361654, A361864, A361866, A361911.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],Median[#]==n/2&]],{n,0,10}]

Formula

a(n) = A079309(floor(n/2)). - Alois P. Heinz, Apr 11 2023