cp's OEIS Frontend

A360687 Number of integer partitions of n whose multiplicities have integer median.

%I A360687 #10 Feb 27 2023 07:46:04
%S A360687 1,2,3,4,5,9,10,16,22,34,42,65,80,115,145,195,240,324,396,519,635,814,
%T A360687 994,1270,1549,1952,2378,2997,3623,4521,5466,6764,8139,10008,12023,
%U A360687 14673,17534,21273,25336,30593,36302,43575,51555,61570,72653,86382,101676
%N A360687 Number of integer partitions of n whose multiplicities have integer median.
%C A360687 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 A360687 The a(1) = 1 through a(8) = 16 partitions:
%e A360687   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A360687        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A360687              (111)  (31)    (41)     (42)      (52)       (53)
%e A360687                     (1111)  (2111)   (51)      (61)       (62)
%e A360687                             (11111)  (222)     (421)      (71)
%e A360687                                      (321)     (2221)     (431)
%e A360687                                      (2211)    (3211)     (521)
%e A360687                                      (3111)    (4111)     (2222)
%e A360687                                      (111111)  (211111)   (3221)
%e A360687                                                (1111111)  (3311)
%e A360687                                                           (4211)
%e A360687                                                           (5111)
%e A360687                                                           (32111)
%e A360687                                                           (221111)
%e A360687                                                           (311111)
%e A360687                                                           (11111111)
%e A360687 For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).
%t A360687 Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]
%Y A360687 The case of an odd number of multiplicities is A090794.
%Y A360687 For mean instead of median we have A360069, ranks A067340.
%Y A360687 These partitions have ranks A360553.
%Y A360687 The complement is counted by A360690, ranks A360554.
%Y A360687 A058398 counts partitions by mean, see also A008284, A327482.
%Y A360687 A124010 gives prime signature, sorted A118914, mean A088529/A088530.
%Y A360687 A325347 = partitions w/ integer median, strict A359907, complement A307683.
%Y A360687 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360687 Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.
%K A360687 nonn
%O A360687 1,2
%A A360687 _Gus Wiseman_, Feb 20 2023