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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
OFFSET
-1,4
COMMENTS
In contrast to A063995 and A105806, here we allow parts that are zeros.
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jan 23 2020
EXTENSIONS
a(35) and beyond from Lars Blomberg, Jan 26 2020
STATUS
approved