A359901 - OEIS

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

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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).

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

Triangle begins:

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.

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

Sequence in context: A221459 A166387 A000209 * A170982 A296339 A362686

Adjacent sequences: A359898 A359899 A359900 * A359902 A359903 A359904

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Jan 21 2023

STATUS

approved