A359893 - OEIS

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.

132

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,10

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..81.

EXAMPLE

Triangle begins:

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

MATHEMATICA

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

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.

Sequence in context: A292251 A231715 A137678 * A218855 A069517 A193526

Adjacent sequences: A359890 A359891 A359892 * A359894 A359895 A359896

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jan 21 2023

STATUS

approved