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(j,k) = rank of the k-th region of the last section of the set of partitions of j.
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
For the definition of "region" see A206437. See also A225600 and A225610. - Omar E. Pol, Aug 12 2013
EXAMPLE
In the triangle T(j,k) for j = 6 the number of regions in the last section of the set of 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 (see below):
From Omar E. Pol, Aug 12 2013: (Start)
---------------------------------------------------------
. Regions Illustration of ranks of the regions
---------------------------------------------------------
. For J=6 k=1 k=2 k=3 k=4
. _ _ _ _ _ _ _ _ _ _ _ _
. |_ _ _ | _ _ _ . |
. |_ _ _|_ | _ _ _ _ * * .| . |
. |_ _ | | _ _ * * . | . |
. |_ _|_ _|_ | * .| .| . |
. | | . |
. | | .|
. | | *|
. | | *|
. | | *|
. | | *|
. |_| *|
.
So row 6 lists: 1 2 2 -5
(End)
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
KEYWORD
sign,tabf
AUTHOR
Omar E. Pol, Dec 04 2011
STATUS
approved