cp's OEIS Frontend

A360686 Number of integer partitions of n whose distinct parts have integer median.

%I A360686 #9 Feb 22 2023 08:08:12
%S A360686 1,2,2,4,3,8,7,16,17,31,35,60,67,99,121,170,200,270,328,436,522,674,
%T A360686 828,1061,1292,1626,1983,2507,3035,3772,4582,5661,6801,8358,10059,
%U A360686 12231,14627,17702,21069,25423,30147,36100,42725,50936,60081,71388,84007,99408
%N A360686 Number of integer partitions of n whose distinct parts have integer median.
%C A360686 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 A360686 The a(1) = 1 through a(8) = 16 partitions:
%e A360686   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A360686        (11)  (111)  (22)    (311)    (33)      (331)      (44)
%e A360686                     (31)    (11111)  (42)      (421)      (53)
%e A360686                     (1111)           (51)      (511)      (62)
%e A360686                                      (222)     (3211)     (71)
%e A360686                                      (321)     (31111)    (422)
%e A360686                                      (3111)    (1111111)  (431)
%e A360686                                      (111111)             (521)
%e A360686                                                           (2222)
%e A360686                                                           (3221)
%e A360686                                                           (3311)
%e A360686                                                           (4211)
%e A360686                                                           (5111)
%e A360686                                                           (32111)
%e A360686                                                           (311111)
%e A360686                                                           (11111111)
%e A360686 For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).
%t A360686 Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Union[#]]]&]],{n,30}]
%Y A360686 For all parts: A325347, strict A359907, ranks A359908, complement A307683.
%Y A360686 For mean instead of median: A360241, ranks A326621.
%Y A360686 These partitions have ranks A360550, complement A360551.
%Y A360686 For multiplicities instead of distinct parts: A360687.
%Y A360686 The complement is counted by A360689.
%Y A360686 A000041 counts integer partitions, strict A000009.
%Y A360686 A000975 counts subsets with integer median.
%Y A360686 A027193 counts odd-length partitions, strict A067659, ranks A026424.
%Y A360686 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A360686 A116608 counts partitions by number of distinct parts.
%Y A360686 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360686 Cf. A240219, A359906, A360005, A360071, A360244, A360245, A360556, A360688.
%K A360686 nonn
%O A360686 1,2
%A A360686 _Gus Wiseman_, Feb 20 2023