|
|
A228350
|
|
Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).
|
|
7
|
|
|
1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 6, 5, 4, 4, 3, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.
The equivalent sequence for integer partitions is A206437.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
---------------------------------------------------------
. Diagram Triangle
Compositions of of compositions (rows)
. of 5 regions and regions (columns)
----------------------------------------------------------
. _ _ _ _ _
. 5 |_ | 5
. 1+4 |_|_ | 1 4
. 2+3 |_ | | 2 3
. 1+1+3 |_|_|_ | 1 1 3
. 3+2 |_ | | 3 2
. 1+2+2 |_|_ | | 1 2 2
. 2+1+2 |_ | | | 2 1 2
. 1+1+1+2 |_|_|_|_ | 1 1 1 2
. 4+1 |_ | | 4 1
. 1+3+1 |_|_ | | 1 3 1
. 2+2+1 |_ | | | 2 2 1
. 1+1+2+1 |_|_|_ | | 1 1 2 1
. 3+1+1 |_ | | | 3 1 1
. 1+2+1+1 |_|_ | | | 1 2 1 1
. 2+1+1+1 |_ | | | | 2 1 1 1
. 1+1+1+1+1 |_|_|_|_|_| 1 1 1 1 1
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
. _ _ _ _
. 4 |_ | 4
. 1+3 |_|_ | 1 3
. 2+2 |_ | | 2 2
. 1+1+2 |_|_|_ | 1 1 2
. 3+1 |_ | | 3 1
. 1+2+1 |_|_ | | 1 2 1
. 2+1+1 |_ | | | 2 1 1
. 1+1+1+1 |_|_|_|_| 1 1 1 1
.
Illustration of the four sections of the set of compositions of 4:
. _ _ _ _
. |_ | 4
. |_|_ | 1+3
. |_ | | 2+2
. _ _ _ |_|_|_ | 1+1+2
. |_ | 3 | | 1
. _ _ |_|_ | 1+2 | | 1
. _ |_ | 2 | | 1 | | 1
. |_| 1 |_| 1 |_| 1 |_| 1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j: 1 2 3 4 5 6 7 8
k
--------------------------------------------------------
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1 |_|1 |_ |2 |_|1 |_ |3 |_|1 |_ |2 |_|1 |_ |4
2 |_|1 |_ |2 |_|1 |_ |3
3 | |1 | |2
4 |_|1 |_ |2
5 | |1
6 | |1
7 | |1
8 |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];
|
|
CROSSREFS
|
Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|