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
Alois P. Heinz, Rows n = 1..145, flattened (first 72 rows from Reinhard Zumkeller)
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.
FORMULA
Sum_{k=-(n-1)..n-1} (-1)^k * T(n,k) = A000025(n). - Alois P. Heinz, Dec 20 2024
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
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