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A063995 Triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k. 16
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 (list; graph; refs; listen; history; text; internal format)
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.

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.

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. 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

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Last modified October 20 20:18 EDT 2019. Contains 328273 sequences. (Running on oeis4.)