A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 4, 3, 2, 0, 0, 0, 0, 1, 7, 4, 3, 1, 0, 0, 0, 0, 1, 8, 6, 3, 2, 0, 0, 0, 0, 0, 1, 12, 8, 4, 3, 1, 0, 0, 0, 0, 0, 1, 14, 11, 5, 4, 2, 0, 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
  0  1
  1  0  1
  1  0  0  1
  2  1  0  0  1
  2  2  0  0  0  1
  4  2  1  0  0  0  1
  4  3  2  0  0  0  0  1
  7  4  3  1  0  0  0  0  1
  8  6  3  2  0  0  0  0  0  1
 12  8  4  3  1  0  0  0  0  0  1
 14 11  5  4  2  0  0  0  0  0  0  1
 21 14  8  4  3  1  0  0  0  0  0  0  1
 24 20 10  5  4  2  0  0  0  0  0  0  0  1
 34 25 15  6  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)
  (3,3,1,1,1)          (6,2,1)      (4,3,2)
  (4,2,1,1,1)          (2,2,2,2,1)  (5,3,1)
  (5,1,1,1,1)          (3,2,2,1,1)
  (2,2,1,1,1,1,1)
  (3,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1)
		

Crossrefs

Column k=1 is A002865(n-1).

Row sums are A027193 (odd-length ptns), strict A067659.

This is the odd-length case of A359901, with half-steps A359893.

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

First appearances of medians are ranked by A360006, A360007.

A000041 counts partitions, strict A000009.

A058398 counts partitions by mean, see also A008284, A327482.

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

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

A325347 counts partitions w/ integer median, complement A307683.

A326567/A326568 gives mean of prime indices.

Cf. A008289, A026424, A327472, A359889, A359895, A359906, A359907, A359910.

Programs

Mathematica

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