A359907 -id:A359907 - cp's OEIS frontend

A360689 Number of integer partitions of n whose distinct parts have non-integer median.

0, 0, 1, 1, 4, 3, 8, 6, 13, 11, 21, 17, 34, 36, 55, 61, 97, 115, 162, 191, 270, 328, 427, 514, 666, 810, 1027, 1211, 1530, 1832, 2260, 2688, 3342, 3952, 4824, 5746, 7010, 8313, 10116, 11915, 14436, 17074, 20536, 24239, 29053, 34170, 40747, 47865, 56830, 66621

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) = 13 partitions:
  .  .  (21)  (211)  (32)    (411)    (43)      (332)      (54)
                     (41)    (2211)   (52)      (611)      (63)
                     (221)   (21111)  (61)      (22211)    (72)
                     (2111)           (322)     (41111)    (81)
                                      (2221)    (221111)   (441)
                                      (4111)    (2111111)  (522)
                                      (22111)              (3222)
                                      (211111)             (6111)
                                                           (22221)
                                                           (222111)
                                                           (411111)
                                                           (2211111)
                                                           (21111111)
For example, the partition y = (5,3,3,2,1,1) has distinct parts {1,2,3,5}, with median 5/2, so y is counted under a(15).

For not just distinct parts: A307683, complement A325347, ranks A359912.
These partitions have ranks A360551.
The complement is counted by A360686, strict A359907, ranks A360550.
For multiplicities instead of distinct parts we have A360690, ranks A360554.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
A360457 gives median of distinct prime indices (times 2).
Cf. A000975, A027193, A090794, A240219, A349156, A360005, A360071, A360241, A360244, A360245.

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

A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 9, 26, 69, 335, 1018, 6629, 22805, 182988, 703745

Offset: 1

Gus Wiseman, Apr 04 2023

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

Since (n+1)/2 is the median of {1..n}, this sequence counts "transitive" set partitions.

			The a(1) = 1 through a(4) = 9 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{23}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with median 5/2, so is counted under a(4).
The set partition {{1,3},{2,4}} has medians {2,3}, with median 5/2, so is counted under a(4).

For mean instead of median we have A361910.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A325347 counts partitions w/ integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361864 counts set partitions with integer median of medians, means A361865.
A361866 counts set partitions with integer sum of medians, means A361911.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A326516, A326521, A326537, A359907, A361654, A361801.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],(n+1)/2==Median[Median/@#]&]],{n,6}]

cp's OEIS Frontend

A360689 Number of integer partitions of n whose distinct parts have non-integer median.

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