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
Examples
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
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.
Crossrefs
Programs
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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 -
Mathematica
Table[ Count[ (First[ # ]-Length[ # ]& /@ IntegerPartitions[ k ]), # ]& /@ Range[ -k+1, k-1 ], {k, 16} ]
Formula
Sum_{k=-(n-1)..n-1} (-1)^k * T(n,k) = A000025(n). - Alois P. Heinz, Dec 20 2024
Extensions
More terms from Vladeta Jovovic and Wouter Meeussen, Sep 19 2001
Comments