A330001 - OEIS

%I #4 Feb 04 2020 14:57:07

%S 0,0,1,1,2,1,4,2,6,6,12,13,25,28,44,54,77,93,127,155,204,247,318,390,

%T 494,610,761,937,1172,1442,1783,2194,2693,3292,4028,4917,5946,7221,

%U 8700,10490,12584,15106,18004,21523,25537,30399,35945,42635,50219,59382

%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.

%F a(n) + A241274(n) + A329976(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 Cf. A000041, A241274, A329976.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Feb 03 2020