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A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank. 6
0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 1, 5, 2, 4, 4, 2, 1, 6, 3, 5, 6, 4, 2, 1, 10, 5, 7, 9, 7, 4, 2, 1, 13, 7, 9, 11, 11, 7, 4, 2, 1, 19, 11, 12, 15, 16, 12, 7, 4, 2, 1, 25, 16, 15, 19, 22, 18, 12, 7, 4, 2, 1, 35, 24, 20, 26, 29, 27, 19, 12, 7, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Every partition of n has n ranks. This is a generalization of the Dyson's rank of a partition which is the largest part minus the number of parts. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1.

It appears that the sum of the k-th ranks of all partitions of n is equal to zero.

Also T(n,k) = number of partitions of n with negative k-th rank.

It appears that reversed rows converge to A000070, the same as A208482. - Omar E. Pol, Mar 11 2012

LINKS

Alois P. Heinz, Rows n = 1..44, flattened

EXAMPLE

For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are

----------------------------------------------------------

Partitions    First      Second       Third      Fourth

of 4          rank        rank        rank        rank

----------------------------------------------------------

4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1

3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0

2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0

2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0

1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1

----------------------------------------------------------

The number of partitions of 4 with positive k-th ranks are 2, 1, 2, 1 so row 4 lists 2, 1, 2, 1.

Triangle begins:

0;

1,   1;

1,   1,  1;

2,   1,  2,  1;

3,   1,  3,  2,  1;

5,   2,  4,  4,  2,  1;

6,   3,  5,  6,  4,  2,  1;

10,  5,  7,  9,  7,  4,  2,  1;

13,  7,  9, 11, 11,  7,  4,  2,  1;

19, 11, 12, 15, 16, 12,  7,  4,  2,  1;

25, 16, 15, 19, 22, 18, 12,  7,  4,  2,  1;

35, 24, 20, 26, 29, 27, 19, 12,  7,  4,  2,  1;

CROSSREFS

Column 1 is A064173. Row sums give A208479.

Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A209616.

Sequence in context: A027353 A027352 A029238 * A274450 A126131 A138012

Adjacent sequences:  A208475 A208476 A208477 * A208479 A208480 A208481

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Mar 07 2012

EXTENSIONS

More terms from Alois P. Heinz, Mar 11 2012

STATUS

approved

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Last modified September 17 22:53 EDT 2019. Contains 327147 sequences. (Running on oeis4.)