cp's OEIS Frontend

A359907 Number of strict integer partitions of n with integer median.

%I A359907 #6 Jan 22 2023 09:14:59
%S A359907 0,1,1,1,2,1,4,2,6,4,9,6,14,10,18,16,27,23,36,34,51,49,67,68,94,95,
%T A359907 122,129,166,174,217,233,287,308,371,405,487,528,622,683,805,880,1024,
%U A359907 1127,1305,1435,1648,1818,2086,2295,2611,2882,3273,3606,4076,4496,5069
%N A359907 Number of strict integer partitions of n with integer median.
%C A359907 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 A359907 The a(1) = 1 through a(14) = 18 partitions (A..E = 10..14):
%e A359907   1  2  3  4   5  6    7    8    9    A    B    C     D     E
%e A359907            31     42   421  53   432  64   542  75    643   86
%e A359907                   51        62   531  73   632  84    652   95
%e A359907                   321       71   621  82   641  93    742   A4
%e A359907                             431       91   731  A2    751   B3
%e A359907                             521       532  821  B1    832   C2
%e A359907                                       541       543   841   D1
%e A359907                                       631       642   931   653
%e A359907                                       721       651   A21   743
%e A359907                                                 732   6421  752
%e A359907                                                 741         761
%e A359907                                                 831         842
%e A359907                                                 921         851
%e A359907                                                 5421        932
%e A359907                                                             941
%e A359907                                                             A31
%e A359907                                                             B21
%e A359907                                                             7421
%t A359907 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[Median[#]]&]],{n,0,30}]
%Y A359907 For mean instead of median: A102627, non-strict A067538 (ranked by A316413).
%Y A359907 This is the strict case of A325347, ranked by A359908.
%Y A359907 The median statistic is ranked by A360005(n)/2.
%Y A359907 A000041 counts partitions, strict A000009.
%Y A359907 A051293 counts subsets with integer mean, median A000975, cf. A005578.
%Y A359907 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359907 A326567/A326568 gives the mean of prime indices.
%Y A359907 A359893, A359901, A359902 count partitions by median.
%Y A359907 Cf. A000016, A082550, A240219, A240850, A326669, A327475, A328966, A359897.
%K A359907 nonn
%O A359907 0,5
%A A359907 _Gus Wiseman_, Jan 21 2023