A330146 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, 0, 1, 1, 3, 4, 7, 9, 13, 16, 24, 29, 39, 51, 69, 87, 118, 152, 199, 256, 330, 418, 534, 670, 838, 1046, 1296, 1603, 1960, 2412, 2936, 3588, 4342, 5288, 6364, 7713, 9272, 11186, 13389, 16117, 19213, 23032, 27408, 32715, 38810, 46176, 54582, 64692, 76286

Offset

0,5

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.

Links

Table of n, a(n) for n=0..48.

Formula

a(n) + A329976(n) = A000041(n) for all n >= 0.

Example

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

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}]

Crossrefs

Cf. A000041, A241274, A329976.

Sequence in context: A004652 A061568 A146994 * A295069 A349360 A103054

Adjacent sequences: A330143 A330144 A330145 * A330147 A330148 A330149

Keyword

nonn,easy

Author

Clark Kimberling, Feb 03 2020

Status

approved