|
| |
|
|
A207034
|
|
Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).
|
|
7
|
|
|
|
0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 8, 8, 9, 9, 9, 10, 6, 7, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 11, 7, 8, 8, 9, 8, 9
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n).
|
|
|
EXAMPLE
|
Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram.
----------------------------------
n Tail a(n)
----------------------------------
15 1 . . . . . . 6
14 1 . . . . . 5
13 1 . . . . . 5
12 1 . . . . 4
11 1 . . . . . 5
10 1 . . . . 4
9 1 . . . . 4
8 1 . . . 3
7 1 . . . . 4
6 1 . . . 3
5 1 . . . 3
4 1 . . 2
3 1 . . 2
2 1 . 1
1 1 0
----------------------------------
Written as a triangle:
0;
1;
2;
2,3;
3,4;
3,4,4,5;
4,5,5,6;
4,5,5,6,6,6,7;
5,6,6,7,6,7,7,8;
5,6,6,7,7,7,8,7,8,8,8,9;
6,7,7,8,7,8,8,9,8,8,9,9,9,10;
6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
...
Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j.
Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix.
---------------------------------------------------------
. j: 1 2 3 4 5 6
n a(n)
---------------------------------------------------------
1 0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 | . 2 . 2 1 . 2 1 1 . 2 1 1 1 . 2 1 1 1 1
3 2 | . . 3 . . 3 1 . . 3 1 1 . . 3 1 1 1
4 2 | . . 2 2 . . 2 2 1 . . 2 2 1 1
5 3 | . . . 4 . . . 4 1 . . . 4 1 1
6 3 | . . . 3 2 . . . 3 2 1
7 4 | . . . . 5 . . . . 5 1
8 3 | . . . 2 2 2
9 4 | . . . . 4 2
10 4 | . . . . 3 3
11 5 | . . . . . 6
...
Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix.
---------------------------------------------------------
. j: 1 2 3 4 5 6
n a(n)
---------------------------------------------------------
1 0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 | 2 . 2 1 . 2 1 1 . 2 1 1 1 . 2 1 1 1 1 .
3 2 | 3 . . 3 1 . . 3 1 1 . . 3 1 1 1 . .
4 2 | 2 2 . . 2 2 1 . . 2 2 1 1 . .
5 3 | 4 . . . 4 1 . . . 4 1 1 . . .
6 3 | 3 2 . . . 3 2 1 . . .
7 4 | 5 . . . . 5 1 . . . .
8 3 | 2 2 2 . . .
9 4 | 4 2 . . . .
10 4 | 3 3 . . . .
11 5 | 6 . . . . .
...
|
|
|
CROSSREFS
|
Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440.
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
