cp's OEIS Frontend

A360689 Number of integer partitions of n whose distinct parts have non-integer median.

%I A360689 #6 Feb 24 2023 21:46:55
%S A360689 0,0,1,1,4,3,8,6,13,11,21,17,34,36,55,61,97,115,162,191,270,328,427,
%T A360689 514,666,810,1027,1211,1530,1832,2260,2688,3342,3952,4824,5746,7010,
%U A360689 8313,10116,11915,14436,17074,20536,24239,29053,34170,40747,47865,56830,66621
%N A360689 Number of integer partitions of n whose distinct parts have non-integer median.
%C A360689 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 A360689 The a(1) = 0 through a(9) = 13 partitions:
%e A360689   .  .  (21)  (211)  (32)    (411)    (43)      (332)      (54)
%e A360689                      (41)    (2211)   (52)      (611)      (63)
%e A360689                      (221)   (21111)  (61)      (22211)    (72)
%e A360689                      (2111)           (322)     (41111)    (81)
%e A360689                                       (2221)    (221111)   (441)
%e A360689                                       (4111)    (2111111)  (522)
%e A360689                                       (22111)              (3222)
%e A360689                                       (211111)             (6111)
%e A360689                                                            (22221)
%e A360689                                                            (222111)
%e A360689                                                            (411111)
%e A360689                                                            (2211111)
%e A360689                                                            (21111111)
%e A360689 For example, the partition y = (5,3,3,2,1,1) has distinct parts {1,2,3,5}, with median 5/2, so y is counted under a(15).
%t A360689 Table[Length[Select[IntegerPartitions[n],!IntegerQ[Median[Union[#]]]&]],{n,30}]
%Y A360689 For not just distinct parts: A307683, complement A325347, ranks A359912.
%Y A360689 These partitions have ranks A360551.
%Y A360689 The complement is counted by A360686, strict A359907, ranks A360550.
%Y A360689 For multiplicities instead of distinct parts we have A360690, ranks A360554.
%Y A360689 A000041 counts integer partitions, strict A000009.
%Y A360689 A116608 counts partitions by number of distinct parts.
%Y A360689 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360689 A360457 gives median of distinct prime indices (times 2).
%Y A360689 Cf. A000975, A027193, A090794, A240219, A349156, A360005, A360071, A360241, A360244, A360245.
%K A360689 nonn
%O A360689 1,5
%A A360689 _Gus Wiseman_, Feb 22 2023