A361801 - OEIS

%I #11 Apr 11 2023 08:39:39

%S 0,0,1,1,4,4,14,14,49,49,175,175,637,637,2353,2353,8788,8788,33098,

%T 33098,125476,125476,478192,478192,1830270,1830270,7030570,7030570,

%U 27088870,27088870,104647630,104647630,405187825,405187825,1571990935,1571990935

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

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

%F a(n) = A079309(floor(n/2)). - _Alois P. Heinz_, Apr 11 2023

%e The subset {1,2,3,5} of {1..5} has median 5/2, so is counted under a(5).

%e The subset {2,3,5} of {1..6} has median 6/2, so is counted under a(6).

%e The a(0) = 0 through a(7) = 14 subsets:

%e   .  .  {1}  {1,2}  {2}      {1,4}      {3}          {1,6}

%e                     {1,3}    {2,3}      {1,5}        {2,5}

%e                     {1,2,3}  {1,2,3,4}  {2,4}        {3,4}

%e                     {1,2,4}  {1,2,3,5}  {1,3,4}      {1,2,5,6}

%e                                         {1,3,5}      {1,2,5,7}

%e                                         {1,3,6}      {1,3,4,5}

%e                                         {2,3,4}      {1,3,4,6}

%e                                         {2,3,5}      {1,3,4,7}

%e                                         {2,3,6}      {2,3,4,5}

%e                                         {1,2,4,5}    {2,3,4,6}

%e                                         {1,2,4,6}    {2,3,4,7}

%e                                         {1,2,3,4,5}  {1,2,3,4,5,6}

%e                                         {1,2,3,4,6}  {1,2,3,4,5,7}

%e                                         {1,2,3,5,6}  {1,2,3,4,6,7}

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

%Y A bisection is A079309.

%Y The case with n's has bisection A057552.

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

%Y A central diagonal of A231147.

%Y A version for partitions is A361849.

%Y For mean instead of median we have A362046.

%Y A000975 counts subsets with integer median, for mean A327475.

%Y A007318 counts subsets by length.

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

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

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

%K nonn,easy

%O 0,5

%A _Gus Wiseman_, Apr 07 2023