A360687 - OEIS

%I #10 Feb 27 2023 07:46:04

%S 1,2,3,4,5,9,10,16,22,34,42,65,80,115,145,195,240,324,396,519,635,814,

%T 994,1270,1549,1952,2378,2997,3623,4521,5466,6764,8139,10008,12023,

%U 14673,17534,21273,25336,30593,36302,43575,51555,61570,72653,86382,101676

%N Number of integer partitions of n whose multiplicities have 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) = 1 through a(8) = 16 partitions:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e        (11)  (21)   (22)    (32)     (33)      (43)       (44)

%e              (111)  (31)    (41)     (42)      (52)       (53)

%e                     (1111)  (2111)   (51)      (61)       (62)

%e                             (11111)  (222)     (421)      (71)

%e                                      (321)     (2221)     (431)

%e                                      (2211)    (3211)     (521)

%e                                      (3111)    (4111)     (2222)

%e                                      (111111)  (211111)   (3221)

%e                                                (1111111)  (3311)

%e                                                           (4211)

%e                                                           (5111)

%e                                                           (32111)

%e                                                           (221111)

%e                                                           (311111)

%e                                                           (11111111)

%e 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 Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

%Y The case of an odd number of multiplicities is A090794.

%Y For mean instead of median we have A360069, ranks A067340.

%Y These partitions have ranks A360553.

%Y The complement is counted by A360690, ranks A360554.

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A124010 gives prime signature, sorted A118914, mean A088529/A088530.

%Y A325347 = partitions w/ integer median, strict A359907, complement A307683.

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

%Y Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 20 2023