1,3

The Integrated Diagram of Partitions is a shell model of partitions of a number. Partitions of n contains all partitions of the previous numbers. The number of shells of the partitions of n is equal to n. The number of parts of the outer shell of the partitions of n is A138137(n)=A006128(n)-A006128(n-1) and equal to the number of terms of row n. The number of terms of row n that are equal to 1 is A000041(n-1). The last term of row n is n. The shell model of partitions has several 2D and 3D versions.

........................................

.. Integrated Diagram of Partitions ...

........... for n = 1 to 9 ............

.......................................

Partition number \ n = 1 2 3 4 5 6 7 8 9

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.1) A000041(1)= 1 .... 1 1 1 1 1 1 1 1 1

.2) A000041(2)= 2 .... . 2 1 1 1 1 1 1 1

.3) A000041(3)= 3 .... . . 3 1 1 1 1 1 1

.4) .................. . 2 . 2 1 1 1 1 1

.5) A000041(4)= 5 .... . . . 4 1 1 1 1 1

.6) .................. . . 3 . 2 1 1 1 1

.7) A000041(5)= 7 .... . . . . 5 1 1 1 1

.8) .................. . 2 . 2 . 2 1 1 1

.9) .................. . . 3 . . 3 1 1 1

10) .................. . . . 4 . 2 1 1 1

11) A000041(6)=11 .... . . . . . 6 1 1 1

12) .................. . . 3 . 2 . 2 1 1

13) .................. . . . 4 . . 3 1 1

14) .................. . . . . 5 . 2 1 1

15) A000041(7)=15 .... . . . . . . 7 1 1

16) .................. . 2 . 2 . 2 . 2 1

17) .................. . . 3 . . 3 . 2 1

18) .................. . . . 4 . 2 . 2 1

19) .................. . . . 4 . . . 4 1

20) .................. . . . . 5 . . 3 1

21) .................. . . . . . 6 . 2 1

22) A000041(8)=22 .... . . . . . . . 8 1

23) .................. . . 3 . 2 . 2 . 2

24) .................. . . 3 . . 3 . . 3

25) .................. . . . 4 . . 3 . 2

26) .................. . . . . 5 . 2 . 2

27) .................. . . . . 5 . . . 4

28) .................. . . . . . 6 . . 3

29) .................. . . . . . . 7 . 2

30) A000041(9)=30 .... . . . . . . . . 9

.......................................

Triangle begins:

1

1,2

1,1,3,

1,1,1,2,2,4

1,1,1,1,1,2,3,5

1,1,1,1,1,1,1,2,2,2,3,3,2,4,6

1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,2,3,4,2,2,5,4,5,3,6,2,7,9

Cf. A000041, A006128, A138137. See A135010 for another version.

Adjacent sequences: A138135 A138136 A138137 this_sequence A138139 A138140 A138141

Sequence in context: A114118 A106177 A135010 this_sequence A080209 A127949 A051340

nonn

Omar E. Pol (info(AT)polprimos.com), Mar 16 2008, Mar 25 2008