login
A359901
Triangle read by rows where T(n,k) is the number of integer partitions of n with median k = 1..n.
97
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
1,7
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).
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[#]==k&]], {n, 15}, {k, n}]
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.
Sequence in context: A221459 A166387 A000209 * A170982 A296339 A362686
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jan 21 2023
STATUS
approved