A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).

1, 0, 1, 1, 2, 2, 6, 6, 10, 13, 20, 26, 39, 50, 71, 87, 121, 156, 208, 265, 348, 440, 566, 712, 906, 1131, 1424, 1766, 2224, 2738, 3390, 4168, 5130, 6266, 7664, 9312, 11332, 13723, 16603, 20004, 24112, 28942, 34708, 41522, 49612, 59031, 70308, 83479, 98992

Offset: 0

Author

Clark Kimberling, Feb 06 2014

Keywords

• nonn

Comments

a(n) + A237365(n) = A000041(n).

Also the number of integer partitions of n whose median difference is 0. For example, the partition (2,2,2,1,1) is counted because its multiset of differences {0,0,0,1} has median 0. - Gus Wiseman, Mar 18 2023

Examples

			Among the 22 partitions of 8, these qualify:  [5,1,1,1], [4,4], [4,1,1,1,1], [3,3,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1], and the remaining 12 do not, so that a(8) = 10.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800

Crossrefs

These partitions have ranks A361204.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts, reverse A058398.

A116608 counts partitions by number of distinct parts.

A359893 and A359901 count partitions by median, odd-length A359902.

Comparing twice the number of distinct parts to the number of parts:

less: A360254, ranks A360558

equal: A239959, ranks A067801

greater: A237365, ranks A361393

less or equal: A237363, ranks A361204

greater or equal: A361394, ranks A361395

Cf. A000975, A027193, A114638, A240219, A325347, A360005, A360244.

Programs

Mathematica

z = 50; t = Map[Length[Select[IntegerPartitions[#], 2*Length[DeleteDuplicates[#]] <= Length[#] &]] &, Range[z]] (*A237363*)
Table[PartitionsP[n] - t[[n]], {n, 1, z}] (*A237365*) (* Peter J. C. Moses, Feb 06 2014 *)
Table[Length[Select[IntegerPartitions[n],Median[Differences[#]]==0&]],{n,0,30}] (* Gus Wiseman, Mar 18 2023 *)