cp's OEIS Frontend

A360244 Number of integer partitions of n where the parts do not have the same median as the distinct parts.

%I A360244 #7 Feb 06 2023 10:06:12
%S A360244 0,0,0,0,1,3,3,9,11,17,23,37,42,68,87,110,153,209,261,352,444,573,750,
%T A360244 949,1187,1508,1909,2367,2938,3662,4507,5576,6826,8359,10203,12372,
%U A360244 15011,18230,21996,26518,31779,38219,45682,54660,65112,77500,92089,109285
%N A360244 Number of integer partitions of n where the parts do not have the same median as the distinct parts.
%C A360244 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 A360244 The a(4) = 1 through a(9) = 17 partitions:
%e A360244   (211)  (221)   (411)    (322)     (332)      (441)
%e A360244          (311)   (3111)   (331)     (422)      (522)
%e A360244          (2111)  (21111)  (511)     (611)      (711)
%e A360244                           (2221)    (4211)     (3222)
%e A360244                           (3211)    (5111)     (3321)
%e A360244                           (4111)    (22211)    (4311)
%e A360244                           (22111)   (32111)    (5211)
%e A360244                           (31111)   (41111)    (6111)
%e A360244                           (211111)  (221111)   (22221)
%e A360244                                     (311111)   (33111)
%e A360244                                     (2111111)  (42111)
%e A360244                                                (51111)
%e A360244                                                (321111)
%e A360244                                                (411111)
%e A360244                                                (2211111)
%e A360244                                                (3111111)
%e A360244                                                (21111111)
%e A360244 For example, the partition y = (33111) has median 1, and the distinct parts {1,3} have median 2, so y is counted under a(9).
%t A360244 Table[Length[Select[IntegerPartitions[n], Median[#]!=Median[Union[#]]&]],{n,0,30}]
%Y A360244 For mean instead of median: A360242, ranks A360246, complement A360243.
%Y A360244 These partitions are ranked by A360248.
%Y A360244 The complement is A360245, ranked by A360249.
%Y A360244 A000041 counts integer partitions, strict A000009.
%Y A360244 A008284 counts partitions by number of parts.
%Y A360244 A116608 counts partitions by number of distinct parts.
%Y A360244 A240219 counts partitions with mean equal to median, ranks A359889.
%Y A360244 A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
%Y A360244 A359893 and A359901 count partitions by median.
%Y A360244 A359894 counts partitions with mean different from median, ranks A359890.
%Y A360244 A360071 counts partitions by number of parts and number of distinct parts.
%Y A360244 Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360068, A360241, A360250, A360251.
%K A360244 nonn
%O A360244 0,6
%A A360244 _Gus Wiseman_, Feb 05 2023