The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A063995 Irregular triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k. 24
 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. 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,  l,  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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 5 16:48 EDT 2021. Contains 346486 sequences. (Running on oeis4.)