A330145 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).

1, 1, 1, 2, 3, 6, 7, 13, 16, 24, 30, 43, 52, 73, 91, 122, 154, 204, 258, 335, 423, 545, 684, 865, 1081, 1348, 1675, 2073, 2546, 3123, 3821, 4648, 5656, 6851, 8282, 9966, 12031, 14416, 17315, 20695, 24754, 29477, 35170, 41738, 49638, 58735, 69613, 82119

Offset: 0

Author

Clark Kimberling, Feb 03 2020

Keywords

• nonn
• easy

Comments

For each partition of n, let

d = number of terms that are not repeated;

r = number of terms that are repeated.

a(n) is the number of partitions such that d >= r.

Examples

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 7
		

Crossrefs

Cf. A000041, A241274, A329976.

Programs

Mathematica

z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] >=  r[p]], {n, 0, z}]

Formula

a(n) + A330001(n) = A000041(n) for all n >= 0.