cp's OEIS Frontend

A360245 Number of integer partitions of n where the parts have the same median as the distinct parts.

%I A360245 #5 Feb 06 2023 10:06:07
%S A360245 1,1,2,3,4,4,8,6,11,13,19,19,35,33,48,66,78,88,124,138,183,219,252,
%T A360245 306,388,450,527,643,780,903,1097,1266,1523,1784,2107,2511,2966,3407,
%U A360245 4019,4667,5559,6364,7492,8601,10063,11634,13469,15469,17985,20558,23812
%N A360245 Number of integer partitions of n where the parts have the same median as the distinct parts.
%C A360245 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 A360245 The a(1) = 1 through a(8) = 11 partitions:
%e A360245   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A360245        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A360245              (111)  (31)    (41)     (42)      (52)       (53)
%e A360245                     (1111)  (11111)  (51)      (61)       (62)
%e A360245                                      (222)     (421)      (71)
%e A360245                                      (321)     (1111111)  (431)
%e A360245                                      (2211)               (521)
%e A360245                                      (111111)             (2222)
%e A360245                                                           (3221)
%e A360245                                                           (3311)
%e A360245                                                           (11111111)
%e A360245 For example, the partition y = (6,4,4,4,1,1) has median 4, and the distinct parts {1,4,6} also have median 4, so y is counted under a(20).
%t A360245 Table[Length[Select[IntegerPartitions[n], Median[#]==Median[Union[#]]&]],{n,0,30}]
%Y A360245 For mean instead of median: A360242, ranks A360247, complement A360243.
%Y A360245 These partitions have ranks A360249.
%Y A360245 The complement is A360244, ranks A360248.
%Y A360245 A000041 counts integer partitions, strict A000009.
%Y A360245 A008284 counts partitions by number of parts.
%Y A360245 A116608 counts partitions by number of distinct parts.
%Y A360245 A240219 counts partitions with mean equal to median, ranks A359889.
%Y A360245 A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
%Y A360245 A359893 and A359901 count partitions by median.
%Y A360245 A359894 counts partitions with mean different from median, ranks A359890.
%Y A360245 A360071 counts partitions by number of parts and number of distinct parts.
%Y A360245 Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360241, A360246, A360250, A360251.
%K A360245 nonn
%O A360245 0,3
%A A360245 _Gus Wiseman_, Feb 05 2023