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partition number




A number that gives the number of ways of placing n indistinguishable balls into n indistinguishable urns. For example:
1:   (*)
2:   (**) (*)(*)
3:   (***) (**)(*) (*)(*)(*)
5:   (****) (***)(*) (**)(**) (**)(*)(*) (*)(*)(*)(*)
7:   (*****) (****)(*) (***)(**) (***)(*)(*) (**)(**)(*) (**)(*)(*)(*) (*)(*)(*)(*)(*)
11: (******) (*****)(*) (****)(**) (****)(*)(*) (***)(***) (***)(**)(*) (***)(*)(*)(*) (**)(**)(**) (**)(**)(*)(*) (**)(*)(*)(*)(*) (*)(*)(*)(*)(*)(*)
The sequence runs: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, ... If the urns are distinguishable, the number of ways is 2n. If the balls are distinguishable, the number of ways is given by the nth Bell number.


Related categories

   • COMBINATORICS
   • TYPES OF NUMBERS