A124943 Table read by rows: number of partitions of n with k as low median.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1, 6, 3, 1, 0, 0, 1, 8, 4, 2, 0, 0, 0, 1, 11, 6, 3, 1, 0, 0, 0, 1, 15, 8, 4, 2, 0, 0, 0, 0, 1, 20, 12, 5, 3, 1, 0, 0, 0, 0, 1, 26, 16, 7, 4, 2, 0, 0, 0, 0, 0, 1, 35, 22, 10, 5, 3, 1, 0, 0, 0, 0, 0, 1, 45, 29, 14, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 58, 40, 19, 8, 5, 3, 1

Offset

1,4

Comments

For a multiset with an odd number of elements, the low median is the same as the median. For a multiset with an even number of elements, the low median is the smaller of the two central elements.

Arrange the parts of a partition nonincreasing order. Remove the first part, then the last, then the first remaining part, then the last remaining part, and continue until only a single number, the low median, remains. - Clark Kimberling, May 16 2019

Links

Table of n, a(n) for n=1..98.

Example

For the partition [2,1^2], the sole middle element is 1, so that is the low median. For [3,2,1^2], the two middle elements are 1 and 2; the low median is the smaller, 1.
First 8 rows:
  1
  1   1
  2   0   1
  3   1   0   1
  4   2   0   0   1
  6   3   1   0   0   1
  8   4   2   0   0   0   1
  11  6   3   1   0   0   0   1
From Gus Wiseman, Jul 09 2023: (Start)
Row n = 8 counts the following partitions:
  (71)        (62)     (53)   (44)  .  .  .  (8)
  (611)       (521)    (431)
  (5111)      (422)    (332)
  (4211)      (3221)
  (41111)     (2222)
  (3311)      (22211)
  (32111)
  (311111)
  (221111)
  (2111111)
  (11111111)
(End)

Mathematica

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 2)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

Crossrefs

Row sums are A000041.

Column k = 1 is A027336, ranks A363488.

The high version of this triangle is A124944.

The rank statistic for this triangle is A363941, high version A363942.

A version for mean instead of median is A363945, rank statistic A363943.

A high version for mean instead of median is A363946, rank stat A363944.

A version for mode instead of median is A363952, high A363953.

A008284 counts partitions by length (or decreasing mean), strict A008289.

A325347 counts partitions with integer median, ranks A359908.

A359893 and A359901 count partitions by median.

A360005(n)/2 returns median of prime indices.

Cf. A025065, A026794, A027193, A067538, A237984, A240219, A362608, A363740.

Sequence in context: A100260 A165317 A174067 * A169803 A099557 A214576

Adjacent sequences: A124940 A124941 A124942 * A124944 A124945 A124946

Keyword

nonn,tabl

Author

Franklin T. Adams-Watters, Nov 13 2006

Status

approved