cp's OEIS Frontend

A329976 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).

%I A329976 #12 Apr 07 2023 08:57:30
%S A329976 0,1,1,2,2,3,4,6,9,14,18,27,38,50,66,89,113,145,186,234,297,374,468,
%T A329976 585,737,912,1140,1407,1758,2153,2668,3254,4007,4855,5946,7170,8705,
%U A329976 10451,12626,15068,18125,21551,25766,30546,36365,42958,50976,60062,70987
%N A329976 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 A329976 For each partition of n, let
%C A329976 d = number of terms that are not repeated;
%C A329976 r = number of terms that are repeated.
%C A329976 a(n) is the number of partitions such that d > r.
%C A329976 Also the number of integer partitions of n with median multiplicity 1. - _Gus Wiseman_, Mar 20 2023
%F A329976 a(n) + A241274(n) + A330001(n) = A000041(n) for n >= 0.
%e A329976 The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
%e A329976 These have d > r:  6, 51, 42, 321
%e A329976 These have d = r:  411, 3222, 21111
%e A329976 These have d < r:  33, 222, 2211, 111111
%e A329976 Thus, a(6) = 4.
%t A329976 z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t A329976 r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] > r[p]], {n, 0, z}]
%Y A329976 For parts instead of multiplicities we have A027336
%Y A329976 The complement is counted by A330001.
%Y A329976 A000041 counts integer partitions, strict A000009.
%Y A329976 A116608 counts partitions by number of distinct parts.
%Y A329976 A237363 counts partitions with median difference 0.
%Y A329976 Cf. A000975, A027193, A067538, A240219, A241274, A325347, A359893, A360005.
%K A329976 nonn,easy
%O A329976 0,4
%A A329976 _Clark Kimberling_, Feb 03 2020