A360689 - OEIS

A360689

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

%I #6 Feb 24 2023 21:46:55

%S 0,0,1,1,4,3,8,6,13,11,21,17,34,36,55,61,97,115,162,191,270,328,427,

%T 514,666,810,1027,1211,1530,1832,2260,2688,3342,3952,4824,5746,7010,

%U 8313,10116,11915,14436,17074,20536,24239,29053,34170,40747,47865,56830,66621

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

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

%e The a(1) = 0 through a(9) = 13 partitions:

%e . . (21) (211) (32) (411) (43) (332) (54)

%e (41) (2211) (52) (611) (63)

%e (221) (21111) (61) (22211) (72)

%e (2111) (322) (41111) (81)

%e (2221) (221111) (441)

%e (4111) (2111111) (522)

%e (22111) (3222)

%e (211111) (6111)

%e (22221)

%e (222111)

%e (411111)

%e (2211111)

%e (21111111)

%e 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).

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

%Y For not just distinct parts: A307683, complement A325347, ranks A359912.

%Y These partitions have ranks A360551.

%Y The complement is counted by A360686, strict A359907, ranks A360550.

%Y For multiplicities instead of distinct parts we have A360690, ranks A360554.

%Y A000041 counts integer partitions, strict A000009.

%Y A116608 counts partitions by number of distinct parts.

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

%Y A360457 gives median of distinct prime indices (times 2).

%Y Cf. A000975, A027193, A090794, A240219, A349156, A360005, A360071, A360241, A360244, A360245.

%K nonn

%O 1,5

%A _Gus Wiseman_, Feb 22 2023