|
|
A228349
|
|
Triangle read by rows: T(j,k) is the k-th part in nondecreasing 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).
|
|
4
|
|
|
1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.
The equivalent sequence for partitions is A220482.
|
|
LINKS
|
|
|
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
.
Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,2,2,3,4;
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
[1];
[1,2];
[1],[1,1,2,3];
[1],[1,2],[1],[1,1,1,1,2,2,3,4];
[1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];
|
|
MATHEMATICA
|
Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* Michael De Vlieger, May 23 2017 *)
|
|
CROSSREFS
|
Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|