A330001 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).

0, 0, 1, 1, 2, 1, 4, 2, 6, 6, 12, 13, 25, 28, 44, 54, 77, 93, 127, 155, 204, 247, 318, 390, 494, 610, 761, 937, 1172, 1442, 1783, 2194, 2693, 3292, 4028, 4917, 5946, 7221, 8700, 10490, 12584, 15106, 18004, 21523, 25537, 30399, 35945, 42635, 50219, 59382

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..49.

Formula

a(n) + A241274(n) + A329976(n) = A000041(n) for 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) = 4.

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: A339241 A007690 A239960 * A292402 A353132 A205685

Adjacent sequences: A329998 A329999 A330000 * A330002 A330003 A330004

Keyword

nonn,easy

Author

Clark Kimberling, Feb 03 2020

Status

approved