A167932 Number of partitions of n such that all parts are equal or all parts are distinct.

1, 1, 2, 3, 4, 4, 7, 6, 9, 10, 13, 13, 20, 19, 25, 30, 36, 39, 51, 55, 69, 79, 92, 105, 129, 144, 168, 195, 227, 257, 303, 341, 395, 451, 515, 588, 676, 761, 867, 985, 1120, 1261, 1433, 1611, 1821, 2053, 2307, 2591, 2919, 3266, 3663, 4100, 4587, 5121, 5725, 6381

Offset

0,3

Comments

Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)).

Links

Table of n, a(n) for n=0..55.

Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)

Omar E. Pol, Illustration of the shell model of partitions (2D view)

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Formula

a(n) = A000041(n) - A167930(n).

a(n) = A000009(n) + A032741(n).

Example

The partitions of 6 are:
6 .............. All parts are distinct ..... (1).
5+1 ............ All parts are distinct ..... (2).
4+2 ............ All parts are distinct ..... (3).
4+1+1 .......... Only some parts are equal.
3+3 ............ All parts are equal ........ (4).
3+2+1 .......... All parts are distinct ..... (5).
3+1+1+1 ........ Only some parts are equal.
2+2+2 .......... All parts are equal ........ (6).
2+2+1+1 ........ Only some parts are equal.
2+1+1+1+1 ...... Only some parts are equal.
1+1+1+1+1+1 .... All parts are equal ........ (7).
So a(6) = 7.

Mathematica

ds[n_]:=Module[{lun=Length[Union[n]]}, Length[n]==lun||lun==1]; Table[ Count[ IntegerPartitions[n], _?(ds)], {n, 0, 60}] (* Harvey P. Dale, Sep 13 2011 *)

Crossrefs

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933.

Sequence in context: A325588 A244903 A342337 * A006087 A364675 A241315

Adjacent sequences: A167929 A167930 A167931 * A167933 A167934 A167935

Keyword

nonn

Author

Omar E. Pol, Nov 15 2009

Extensions

More terms from D. S. McNeil, May 10 2010

Status

approved