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

Author

Omar E. Pol, Nov 15 2009

Keywords

• nonn

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)).

Examples

			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.
		

Links

• 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)

Crossrefs

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

Programs

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 *)

Formula

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

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

Extensions

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