|
| |
|
|
A135010
|
|
A shell model of partitions. Triangle read by rows in which row n lists the parts of the outer shell of the partitions of n.
|
|
137
| |
|
|
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, 2, 4, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 5, 3, 4, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 2, 3, 3, 2, 6, 3, 5, 4, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
COMMENTS
| This is the original sequence of a large number of sequences connected with the shell model of partitions.
The set of partitions of n contains all partitions of the previous numbers. Each part of a partition belonging to a shell or subshell. 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 shell model of partitions has several 2D and 3D models. In this sequence the last term of row n is n.
The shell model of partitions can be interpreted as a table of partitions. See also A138121. - Omar E. Pol, Nov 18 2009
It appears that the versions of the model show an overlapping of shells and subshells of the numbers congruent to k mod m into parts >= m. For example:
First generation (the main table):
Table 1.0: Partitions of integers congruent to 0 mod 1 into parts >= 1.
Second generation:
Table 2.0: Partitions of integers congruent to 0 mod 2 into parts >= 2.
Table 2.1: Partitions of integers congruent to 1 mod 2 into parts >= 2.
Third generation:
Table 3.0: Partitions of integers congruent to 0 mod 3 into parts >= 3.
Table 3.1: Partitions of integers congruent to 1 mod 3 into parts >= 3.
Table 3.2: Partitions of integers congruent to 2 mod 3 into parts >= 3.
And so on.
Conjecture:
Let j and n be integers congruent to k mod m such that 0<=k<m<=j<n. Let h=(n-j)/m. Consider only all partitions of n into parts >= m. Then remove every partition in which the parts of size m appears a number of times < h. Then remove h parts of size m in every partition. The rest are the partitions of j into parts >= m. (Note that in the shell model, h is the number of shells or subshells removed), (Omar E. Pol, Dec 05 2010, Dec 06 2010).
|
|
|
LINKS
| O. E. Pol, Illustration of the shell model of partitions [From Omar E. Pol, Nov 18 2009]
O. E. Pol, Illustration of the shell model of partitions (2D view) [From Omar E. Pol, Nov 18 2009]
O. E. Pol, Illustration of the shell model of partitions (3D view) [From Omar E. Pol, Nov 18 2009]
|
|
|
EXAMPLE
| Triangle begins:
1
1,2
1,1,3
1,1,1,2,2,4
1,1,1,1,1,2,3,5
1,1,1,1,1,1,1,2,2,2,2,4,3,3,6
1,1,1,1,1,1,1,1,1,1,1,2,2,3,2,5,3,4,7
...
=========================================================
. The Shell Model of Partitions version "Tree"
. (Three views of the first seven shells)
---------------------------------------------------------
. Table 1.0
j A000041 A194550 A135010
=========================================================
1 p(1) = 1 1 1 1 1 1 1 1 1
2 p(2) = 2 2 1 2 . 1 1 1 1 1
3 p(3) = 3 1 3 3 . . 1 1 1 1
4 2 1 2 . 2 . 1 1 1
5 p(4) = 5 4 1 4 . . . 1 1 1
6 1 3 3 . . 2 . 1 1
7 p(5) = 7 1 5 5 . . . . 1 1
8 2 1 2 . 2 . 2 . 1
9 4 1 4 . . . 2 . 1
10 3 1 3 . . 3 . . 1
11 p(6) = 11 6 1 6 . . . . . 1
12 . 3 3 . . 2 . 2 .
13 . 5 5 . . . . 2 .
14 . 4 4 . . . 3 . .
15 p(7) = 15 . 7 7 . . . . . .
. .
. A182730 A182731 A141285
.
. A182746 <- 4 . 2 1 0 1 2 . 4 -> A182747
---------------------------------------------------------
.
. A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
. . . . . 1 . . . .
. . . . 2 1 . . . .
. Table 2.0 . 3 . . 1 2 . . . Table 2.1
. . . 2 2 1 . . 3 .
. . . . . 1 2 2 . .
. 1 . . . .
. A182742 A182982 A182743 A182983
. A182992 A182994 A182993 A182995
.
Each column contains the same parts. Each part can be represented by a cuboid of size 1 x 1 x L, where L is the size of the part.
|
|
|
CROSSREFS
| Cf. A000041, A006128, A026791, A026792, A138121, A138135, A138137, A138879, A138880, A141285, A144120, A144300, A167928. - Omar E. Pol, Nov 18 2009
Cf. A182718, A182730 - A182733, A182742, A182743, A182746, A182747, A182982, A182983.- Omar E. Pol, Jan 24 2011
Sequence in context: A146014 A202241 A106177 * A138138 A196931 A175465
Adjacent sequences: A135007 A135008 A135009 * A135011 A135012 A135013
|
|
|
KEYWORD
| nonn,tabf
|
|
|
AUTHOR
| Omar E. Pol (info(AT)polprimos.com), Nov 17 2007, Nov 18 2007, Mar 21 2008
|
| |
|
|
