A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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

Offset: 1

Author

Omar E. Pol, Mar 16 2008, Mar 25 2008

Keywords

• nonn
• tabf
• less

Comments

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 last section of the set of 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.

Examples

			........................................
.. Integrated Diagram of Partitions ...
........... for n = 1 to 9 ............
.......................................
Partition number \ n = 1 2 3 4 5 6 7 8 9
........................................
.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
		

Links

• Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.

Crossrefs

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

Programs

Mathematica

Table[ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ Reverse@Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)