A194447 Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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

Offset: 1

Author

Omar E. Pol, Dec 04 2011

Keywords

• sign
• tabf

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

Examples

			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

Row j has length A187219(j). The absolute value of the last term of row j is A000094(j+1). Row sums give A000004.

Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A206437.

Formula

a(n) = A141285(n) - A194446(n). - Omar E. Pol, Dec 05 2011