A063995 Irregular triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k.

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2

Offset

1,30

Comments

The rank of a partition is the largest part minus the number of parts.

The rows are symmetric: for every partition of rank r there is its conjugate with rank -r. [Joerg Arndt, Oct 07 2012]

Links

Reinhard Zumkeller, Rows n = 1..72 of triangle, flattened

G. E. Andrews, The number of smallest parts in the partitions of n. [Also Selected Works, p. 603, see N(m,n).] - N. J. A. Sloane, Dec 16 2013

A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4, (1954). 84-106. Math. Rev. 15,685d.

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 1.

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

Example

The partition 5 = 4+1 has largest summand 4 and 2 summands, hence has rank 4-2 = 2.
Triangle begins:
[ 1]                               1,
[ 2]                            1, 0, 1,
[ 3]                         1, 0, 1, 0, 1,
[ 4]                      1, 0, 1, 1, 1, 0, 1,
[ 5]                   1, 0, 1, 1, 1, 1, 1, 0, 1,
[ 6]                1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1,
[ 7]             1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1,
[ 8]          1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1,
[ 9]       1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1,
[10]    1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1,
[11] 1, 0, 1, 1, 2, ...
Row 20 is:
T(20, k) = 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 22, 30, 33, 40, 42, 48, 45, 48, 42, 40, 33, 30, 22, 20, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1; -19 <= k <= 19.
Another view of the table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):
n\m -6 -5  -4  -3  -2  -1   0   1   2   3   4   5   6
-----------------------------------------------------
0   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
1   0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,
2   0,  0,  0,  0,  0,  1,  0,  1,  0,  0,  0,  0,  0,
3   0,  0,  0,  0,  1,  0,  1,  0,  1,  0,  0,  0,  0,
4   0,  0,  0,  1,  0,  1,  1,  1,  0,  1,  0,  0,  0,
5   0,  0,  1,  0,  1,  1,  1,  1,  1,  0,  1,  0,  0,
6   0,  1,  0,  1,  1,  2,  1,  2,  1,  1,  0,  1,  0,
7   1,  0,  1,  1,  2,  1,  3,  1,  2,  1,  1,  0,  1,
...
The central triangle is the present sequence, the right-hand triangle is A105806. - N. J. A. Sloane, Jan 23 2020

Mathematica

Table[ Count[ (First[ # ]-Length[ # ]& /@ IntegerPartitions[ k ]), # ]& /@ Range[ -k+1, k-1 ], {k, 16} ]

Prog

(Haskell)
import Data.List (sort, group)
a063995 n k = a063995_tabf !! (n-1) !! (n-1+k)
a063995_row n = a063995_tabf !! (n-1)
a063995_tabf = [[1], [1, 0, 1]] ++ (map
   (\rs -> [1, 0] ++ (init $ tail $ rs) ++ [0, 1]) $ drop 2 $ map
   (map length . group . sort . map rank) $ tail pss) where
      rank ps = maximum ps - length ps
      pss = [] : map (\u -> [u] : [v : ps | v <- [1..u],
                             ps <- pss !! (u - v), v <= head ps]) [1..]
-- Reinhard Zumkeller, Jul 24 2013

Crossrefs

For the number of partitions of n with rank 0 (balanced partitions) see A047993.

Cf. A105806 (right half of triangle), A005408 (row lengths), A000041 (row sums), A047993 (central terms).

Sequence in context: A052308 A116510 A128915 * A280737 A322305 A020951

Adjacent sequences: A063992 A063993 A063994 * A063996 A063997 A063998

Keyword

nonn,nice,tabf

Author

N. J. A. Sloane, Sep 19 2001

Extensions

More terms from Vladeta Jovovic and Wouter Meeussen, Sep 19 2001

Status

approved