login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank. 12
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.

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

From Omar E. Pol, Dec 12 2019: (Start)

1) The k-th part of a partition of n is also the number of parts >= k of its conjugate partition.

2) The k-th rank of a partitions is also the number of parts >= k of its conjugate partition minus the number of parts >= k.

For example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. The mentioned conjugate partition is [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. the Third part is 2, so the conjugate partition has two parts >= 3, and so on. In this case the conjugate partition is [5, 3, 1].

3) The difference between the k-th part and the (k+1)-st part of the partition of n is also the number of k's in its conjugate partition. For example: consider the partition [5, 3, 1]. The difference between the first and the second part is 5 - 3 = 2, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 3 - 1 = 2, equals the number of 2's in its conjugate partition. The difference between the third and the fourth (virtual) part is 1 - 0 = 1, equals the number of 3's in its conjugate partition [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The difference between the first and the second part is 3 - 2 = 1, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 2 - 2 = 0, equals the number of 2's in its conjugate partition. The difference between the third and the fourth part is 2 - 1 = 1, equals the number of 3's in its conjugate partition, and so on.

4) The list of n ranks of a partition of n equals the list of n ranks multiplied by -1 of its conjugate partition. For example the nine ranks of the partition [5, 3, 1] of 9 are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0].

For a list of partitions of the positive integers ordered by its k-th ranks see A330370. (End)

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, A208483, A209616, A330368, A330369, A330370.

Sequence in context: A027352 A029238 A345137 * A274450 A126131 A343332

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 22:25 EST 2021. Contains 349567 sequences. (Running on oeis4.)