A167930 Number of partitions of n in which some but not all parts are equal.

0, 0, 0, 0, 1, 3, 4, 9, 13, 20, 29, 43, 57, 82, 110, 146, 195, 258, 334, 435, 558, 713, 910, 1150, 1446, 1814, 2268, 2815, 3491, 4308, 5301, 6501, 7954, 9692, 11795, 14295, 17301, 20876, 25148, 30200, 36218, 43322, 51741, 61650, 73354

Offset

0,6

Comments

The parts may not all be equal, and at least one part must occur at least twice. - N. J. A. Sloane, May 30 2024

Links

Table of n, a(n) for n=0..44.

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) = A047967(n) - A032741(n).

a(n) = A000041(n) - A000009(n) - A032741(n).

a(0) = 0: For n>0, a(n) = A000041(n) - A000009(n) - A000005(n) + 1.

Example

The partitions of 6 are:
6 ....................... All parts are distinct.
5 + 1 ................... All parts are distinct.
4 + 2 ................... All parts are distinct.
4 + 1 + 1 ............... Only some parts are equal ...... (1).
3 + 3 ................... All parts are equal.
3 + 2 + 1 ............... All parts are distinct.
3 + 1 + 1 + 1 ........... Only some parts are equal ...... (2).
2 + 2 + 2 ............... All parts are equal.
2 + 2 + 1 + 1 ........... Only some parts are equal ...... (3).
2 + 1 + 1 + 1 + 1 ....... Only some parts are equal ...... (4).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
Then a(6) = 4.
a(7) = 9 from 511  4111  331  322  3211  31111  2221  22111  211111. - N. J. A. Sloane, May 30 2024

Mathematica

f[lst_]:=With[{c=Split[lst]}, Length[lst]>2&&Max[Length/@c]>1&&Length[c]>1]; Table[Length[ Select[ IntegerPartitions[n], f]], {n, 0, 50}] (* Harvey P. Dale, May 30 2024 *)

Crossrefs

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167931, A167932, A167933.

Sequence in context: A339996 A225288 A377040 * A326981 A124285 A131326

Adjacent sequences: A167927 A167928 A167929 * A167931 A167932 A167933

Keyword

nonn

Author

Omar E. Pol, Nov 15 2009

Extensions

Edited by Omar E. Pol, Nov 16 2009

More terms from Max Alekseyev, May 02 2011

Status

approved