A124944 Table, number of partitions of n with k as high median.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3

Offset: 1

Author

Franklin T. Adams-Watters, Nov 13 2006

Keywords

• nonn
• tabl

Comments

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

This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017

Arrange the parts of a partition nonincreasing order. Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019

Examples

			For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
From _Gus Wiseman_, Jul 12 2023: (Start)
Triangle begins:
   1
   1  1
   1  1  1
   2  1  1  1
   3  1  1  1  1
   4  3  1  1  1  1
   6  4  1  1  1  1  1
   8  6  3  1  1  1  1  1
  11  8  5  1  1  1  1  1  1
  15 11  7  3  1  1  1  1  1  1
  20 15  9  5  1  1  1  1  1  1  1
  26 21 12  8  3  1  1  1  1  1  1  1
  35 27 16 10  5  1  1  1  1  1  1  1  1
  45 37 21 13  8  3  1  1  1  1  1  1  1  1
  58 48 29 16 11  5  1  1  1  1  1  1  1  1  1
Row n = 8 counts the following partitions:
  (611)       (521)    (431)   (44)  (53)  (62)  (71)  (8)
  (5111)      (422)    (332)
  (41111)     (4211)   (3311)
  (32111)     (3221)
  (311111)    (2222)
  (221111)    (22211)
  (2111111)
  (11111111)
(End)
		

Crossrefs

Row sums are A000041.

Column k = 1 is A027336(n-1), ranks A364056.

Column k = 1 in the low version is A027336, ranks A363488.

The low version of this triangle is A124943.

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

A version for mean instead of median is A363946, low A363945.

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

A008284 counts partitions by length, maximum, or decreasing mean.

A026794 counts partitions by minimum, strict A026821.

A325347 counts partitions with integer median, ranks A359908.

A359893 and A359901 count partitions by median.

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

Cf. A008289, A025065, A027193, A067538, A237984, A240219, A362608, A363740, A363943, A363944.

Programs

Mathematica

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