A331447 Triangle read by rows: T(n,k) (n >= 0, -1 <= k <= n-1) = number of partitions of n into nonnegative integer parts with rank k.

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 1, 1, 8, 6, 5, 3, 2, 1, 1, 10, 9, 6, 5, 3, 2, 1, 1, 15, 12, 10, 7, 5, 3, 2, 1, 1, 20, 17, 13, 10, 7, 5, 3, 2, 1, 1, 28, 23, 19, 14, 11, 7, 5, 3, 2, 1, 1, 36, 31, 25, 20, 14, 11, 7, 5, 3, 2, 1, 1, 50, 42

Offset

-1,4

Comments

In contrast to A063995 and A105806, here we allow parts that are zeros.

Links

Lars Blomberg, Table of n, a(n) for n = -1..5048

Alexander Berkovich and Frank G. Garvan, Some observations on Dyson's new symmetries of partitions, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 2.

Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180.

Formula

See Dyson (1969).

Example

Triangle begins:
   1,
   1, 1,
   2, 1, 1,
   2, 2, 1, 1,
   4, 3, 2, 1, 1,
   5, 4, 3, 2, 1, 1,
   8, 6, 5, 3, 2, 1, 1,
  10, 9, 6, 5, 3, 2, 1, 1,
  ...
If we include negative values of the rank k, we get the following table, taken from Dyson (1969):
  n\k| -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6
  ---+---------------------------------------------------
   0 |  1,  1,  1,  1,  1,  1,  0,  0,  0,  0,  0,  0,  0, ...
   1 |  1,  1,  1,  1,  1,  1,  1,  0,  0,  0,  0,  0,  0, ...
   2 |  2,  2,  2,  2,  2,  2,  1,  1,  0,  0,  0,  0,  0, ...
   3 |  3,  3,  3,  3,  3,  2,  2,  1,  1,  0,  0,  0,  0, ...
   4 |  5,  5,  5,  5,  4,  4,  3,  2,  1,  1,  0,  0,  0, ...
   5 |  7,  7,  7,  6,  6,  5,  4,  3,  2,  1,  1,  0,  0, ...
   6 | 11, 11, 10, 10,  9,  8,  6,  5,  3,  2,  1,  1,  0, ...
   7 | 15, 14, 14, 13, 12, 10,  9,  6,  5,  3,  2,  1,  1, ...
   ...
Starting at column k=-1 gives the present triangle.

Crossrefs

For the rank of a partition see A063995, A105806.

Sequence in context: A240656 A106476 A101566 * A352460 A342767 A176653

Adjacent sequences: A331444 A331445 A331446 * A331448 A331449 A331450

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 23 2020

Extensions

a(35) and beyond from Lars Blomberg, Jan 26 2020

Status

approved