|
| |
|
|
A194447
|
|
Rank of the n-th region of the shell model of partitions.
|
|
17
|
|
|
|
0, 0, 0, 1, -1, 2, -2, 1, 2, 2, -5, 2, 3, 3, -8, 1, 2, 2, 2, 4, 3, -14, 2, 3, 3, 3, 2, 4, 4, -21, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -32, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -45, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -65
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,6
|
|
|
COMMENTS
|
Here the rank of a region is defined to be the largest part minus the number of parts. The same idea as the Dyson's rank of a partition.
Also triangle read by rows: T(n,k) = rank of the k-th region of the outer shell of the partitions of n.
Conjecture: the sum of every row is equal to zero.
Note that in some rows there are several negative terms. - Omar E. Pol, Oct 27 2012
|
|
|
LINKS
|
Table of n, a(n) for n=1..77.
|
|
|
FORMULA
|
a(n) = A141285(n) - A194446(n). - Omar E. Pol, Dec 05 2011
|
|
|
EXAMPLE
|
For n = 6 the number of regions in the outer shell of the partitions of 6 is equal to 4. The first region given by 2 has rank 2-1 = 1. The second region given by 4+2 has rank 4-2 = 2. The third region given by 3 has rank 3-1 = 2. The fourth region given by 6+3+2+2+1+1+1+1+1+1+1 has rank 6-11 = -5.
Written as a triangle begins:
0;
0;
0;
1,-1;
2,-2;
1,2,2,-5;
2,3,3,-8;
1,2,2,2,4,3,-14;
2,3,3,3,2,4,4,-21;
1,2,2,2,4,3,1,3,5,5,4,-32;
2,3,3,3,2,4,4,1,4,3,5,6,5,-45;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-65;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3,3,5,5,4,5,4,7,7,6,-88;
|
|
|
CROSSREFS
|
Row n has length A187219(n). The absolute value of the last term of row n is A000094(n+1). It appears that row sums give A000004.
Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446.
Sequence in context: A156748 A075445 A216612 * A077766 A002345 A071694
Adjacent sequences: A194444 A194445 A194446 * A194448 A194449 A194450
|
|
|
KEYWORD
|
sign,tabf
|
|
|
AUTHOR
|
Omar E. Pol, Dec 04 2011
|
|
|
STATUS
|
approved
|
| |
|
|
