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%I #12 Apr 07 2023 08:57:30
%S 0,1,1,2,2,3,4,6,9,14,18,27,38,50,66,89,113,145,186,234,297,374,468,
%T 585,737,912,1140,1407,1758,2153,2668,3254,4007,4855,5946,7170,8705,
%U 10451,12626,15068,18125,21551,25766,30546,36365,42958,50976,60062,70987
%N Number of partitions p of n such that (number of numbers in p that have multiplicity 1) > (number of numbers in p having multiplicity > 1).
%C For each partition of n, let
%C d = number of terms that are not repeated;
%C r = number of terms that are repeated.
%C a(n) is the number of partitions such that d > r.
%C Also the number of integer partitions of n with median multiplicity 1. - _Gus Wiseman_, Mar 20 2023
%F a(n) + A241274(n) + A330001(n) = A000041(n) for n >= 0.
%e The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
%e These have d > r: 6, 51, 42, 321
%e These have d = r: 411, 3222, 21111
%e These have d < r: 33, 222, 2211, 111111
%e Thus, a(6) = 4.
%t z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] > r[p]], {n, 0, z}]
%Y For parts instead of multiplicities we have A027336
%Y The complement is counted by A330001.
%Y A000041 counts integer partitions, strict A000009.
%Y A116608 counts partitions by number of distinct parts.
%Y A237363 counts partitions with median difference 0.
%Y Cf. A000975, A027193, A067538, A240219, A241274, A325347, A359893, A360005.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Feb 03 2020
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