A359893 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.

1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 3, 0, 1, 2, 0, 0, 0, 0, 1, 4, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 6, 1, 3, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 8, 1, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 11, 2, 7, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gus Wiseman, Jan 21 2023

Keywords

• nonn
• tabf

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

			Triangle begins:
  1
  1  0  1
  1  1  0  0  1
  2  0  2  0  0  0  1
  3  0  1  2  0  0  0  0  1
  4  1  2  0  3  0  0  0  0  0  1
  6  1  3  0  1  3  0  0  0  0  0  0  1
  8  1  6  0  2  0  4  0  0  0  0  0  0  0  1
 11  2  7  1  3  0  1  4  0  0  0  0  0  0  0  0  1
 15  2 10  3  4  0  2  0  5  0  0  0  0  0  0  0  0  0  1
 20  3 13  3  7  0  3  0  1  5  0  0  0  0  0  0  0  0  0  0  1
 26  4 19  3 11  1  4  0  2  0  6  0  0  0  0  0  0  0  0  0  0  0  1
For example, row n = 8 counts the following partitions:
  611       4211  422    .  332  .  44  .  .  .  .  .  .  .  8
  5111            521       431     53
  32111           2222              62
  41111           3221              71
  221111          3311
  311111          22211
  2111111
  11111111
		

Crossrefs

Row sums are A000041.

Row lengths are 2n-1 = A005408(n-1).

Column k=1 is A027336(n+1).

For mean instead of median we have A058398, see also A008284, A327482.

The mean statistic is ranked by A326567/A326568.

Omitting half-steps gives A359901.

The odd-length case is A359902.

The median statistic is ranked by A360005(n)/2.

First appearances of medians are ranked by A360006, A360007.

A027193 counts odd-length partitions, strict A067659, ranked by A026424.

A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.

A240219 counts partitions w/ the same mean as median, complement A359894.

Cf. A325347, A349156, A359889, A359895, A359906, A359907, A360008.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Median[#]==k&]],{n,1,10},{k,1,n,1/2}]