A359901 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k = 1..n.

1, 1, 1, 1, 0, 1, 2, 2, 0, 1, 3, 1, 0, 0, 1, 4, 2, 3, 0, 0, 1, 6, 3, 1, 0, 0, 0, 1, 8, 6, 2, 4, 0, 0, 0, 1, 11, 7, 3, 1, 0, 0, 0, 0, 1, 15, 10, 4, 2, 5, 0, 0, 0, 0, 1, 20, 13, 7, 3, 1, 0, 0, 0, 0, 0, 1, 26, 19, 11, 4, 2, 6, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gus Wiseman, Jan 21 2023

Keywords

• nonn
• tabl

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  1
   1  0  1
   2  2  0  1
   3  1  0  0  1
   4  2  3  0  0  1
   6  3  1  0  0  0  1
   8  6  2  4  0  0  0  1
  11  7  3  1  0  0  0  0  1
  15 10  4  2  5  0  0  0  0  1
  20 13  7  3  1  0  0  0  0  0  1
  26 19 11  4  2  6  0  0  0  0  0  1
  35 24 14  5  3  1  0  0  0  0  0  0  1
  45 34 17  8  4  2  7  0  0  0  0  0  0  1
  58 42 23 12  5  3  1  0  0  0  0  0  0  0  1
For example, row n = 9 counts the following partitions:
  (7,1,1)              (5,2,2)      (3,3,3)  (4,4,1)  .  .  .  .  (9)
  (6,1,1,1)            (6,2,1)      (4,3,2)
  (3,3,1,1,1)          (3,2,2,2)    (5,3,1)
  (4,2,1,1,1)          (4,2,2,1)
  (5,1,1,1,1)          (4,3,1,1)
  (3,2,1,1,1,1)        (2,2,2,2,1)
  (4,1,1,1,1,1)        (3,2,2,1,1)
  (2,2,1,1,1,1,1)
  (3,1,1,1,1,1,1)
  (2,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1)
		

Crossrefs

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

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

Row sums are A325347.

The mean statistic is ranked by A326567/A326568.

Including half-steps gives A359893.

The odd-length case is A359902.

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

First appearances of medians are ranked by A360006, A360007.

A000041 counts partitions, strict A000009.

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

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

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

Cf. A008289, A237984, A327472, A349156, A359889, A359907.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Median[#]==k&]],{n,15},{k,n}]