A360069 -id:A360069 - cp's OEIS frontend

A360690 Number of integer partitions of n with non-integer median of multiplicities.

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Gus Wiseman, Feb 22 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272

Offset: 1

Gus Wiseman, Jan 27 2023

nonn

Conjecture: No term > 1 is squarefree.

			A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.

Positions of positive terms in A360068, ranked by A359903.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
Cf. A005117, A067340, A067538, A240219, A316313, A349156, A359904, A359905, A360069.

Select[Range[30],Select[IntegerPartitions[#],Mean[#]==Mean[Length/@Split[#]]&]!={}&]

a(22)-a(58) from Alois P. Heinz, Jan 29 2023

cp's OEIS Frontend

A360690 Number of integer partitions of n with non-integer median of multiplicities.

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