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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, 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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 27 11:17 EDT 2022. Contains 354896 sequences. (Running on oeis4.)