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

0,5

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).

Links

Table of n, a(n) for n=0..35.

Formula

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

Example

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}

Mathematica

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

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.

Sequence in context: A255297 A339319 A347428 * A263870 A263796 A263871

Adjacent sequences: A361798 A361799 A361800 * A361802 A361803 A361804

Keyword

nonn,easy

Author

Gus Wiseman, Apr 07 2023

Status

approved