A261036 Table read by rows: number of complete partitions of n with largest part = k.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 2, 2, 1, 3, 4, 2, 1, 4, 5, 4, 2, 1, 4, 6, 5, 4, 1, 5, 8, 8, 5, 4, 1, 5, 10, 10, 8, 5, 1, 6, 11, 14, 10, 8, 5, 1, 6, 14, 16, 16, 10, 8, 1, 7, 16, 22, 20, 16, 10, 8, 1, 7, 18, 26, 27, 20, 16, 10, 1, 8, 21, 32, 34, 31

Offset: 1

Author

Reinhard Zumkeller, Aug 08 2015

Keywords

• nonn
• tabf
• look

Comments

See A126796 for definition of complete partitions;

A126796(n) = sum of n-th row;

also T(n,floor((n+1)/2)) = A126796(floor(n/2)).

Examples

			T(8,2) = #{1+1+1+1+1+1+2, 1+1+1+1+2+2, 1+1+2+2+2} = 3;
T(8,3) = #{1+1+1+1+1+3, 1+1+1+2+3, 1+1+3+3, 1+2+2+3} = 4;
T(8,4) = #{1+1+1+1+4, 1+1+2+4} = 2;
T(9,2) = #{+11+1+1+1+1+1+2, 1+1+1+1+1+2+2, 1+1+1+2+2+2, 1+2+2+2+2} = 4;
T(9,3) = #{1+1+1+1+1+1+3, 1+1+1+1+2+3, 1+1+1+3+3, 1+1+2+2+3, 3,3,2,1} = 5;
T(9,4) = #{1+1+1+1+1+4, 1+1+1+2+4, 1+1+3+4, 1+2+2+4} = 4;
T(9,5) = #{1+1+1+1+5, 1+1+2+2+5} = 2.
. -----------------------------------------------
.   n |  T(n,k), k = 1 .. [(n+1)/2]  | A126796(n)
. ----+------------------------------+-----------
.   1 |  1                           |         1
.   2 |  1                           |         1
.   3 |  1 1                         |         2
.   4 |  1 1                         |         2
.   5 |  1 2  1                      |         4
.   6 |  1 2  2                      |         5
.   7 |  1 3  2  2                   |         8
.   8 |  1 3  4  2                   |        10
.   9 |  1 4  5  4  2                |        16
.  10 |  1 4  6  5  4                |        20
.  11 |  1 5  8  8  5  4             |        31
.  12 |  1 5 10 10  8  5             |        39
.  13 |  1 6 11 14 10  8  5          |        55
.  14 |  1 6 14 16 16 10  8          |        71
.  15 |  1 7 16 22 20 16 10  8       |       100
.  16 |  1 7 18 26 27 20 16 10       |       125
.  17 |  1 8 21 32 34 31 20 16 10    |       173
.  18 |  1 8 24 37 42 39 31 20 16    |       218
.  19 |  1 9 26 46 53 50 39 31 20 16 |       291
.  20 |  1 9 30 52 66 63 55 39 31 20 |       366
		

Links

• Reinhard Zumkeller, Rows n = 1..200 of triangle, flattened
• SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

Crossrefs

Cf. A008619 (row lengths), A126796 (row sums).

Cf. A122197.

Programs

Haskell

import Data.MemoCombinators (memo2, integral, Memo)
a261036 n k = a261036_tabf !! (n-1) !! (k-1)
a261036_row n = a261036_tabf !! (n-1)
a261036_tabf = zipWith (map . flip dMemo) [1..] a122197_tabf where
   dMemo = memo2 integral integral d
   d 0 _ = 0
   d _ 0 = 0
   d 1 _ = 1
   d k n | n <= 2 * k - 2 = 0
         | n <= 3 * k - 2 = dMemo (k - 1) (n - 1)
         | otherwise      = dMemo (k - 1) (n - 1) + dMemo k (n - k)

Mathematica

d[k_, n_] := d[k, n] = Which[n == 0 || k == 0, 0, k == 1, 1, n >= 3 k - 1, d[k - 1, n - 1] + d[k, n - k], 2 k - 1 <= n <= 3 k - 2, d[k - 1, n - 1], True, 0]; Table[d[k, n], {n, 17}, {k, Floor[(n + 1)/2]}] // Flatten (* Michael De Vlieger, Jul 13 2017 *)

Formula

According to the Park link, Theorem 3.7, p. 357f:

Let D_k(n) be the number of complete partitions of a positive integer n with largest part exactly k.

D_0(n) = 0 for all n, D_k(0) = 0 for all k, D_1(n)=1 for n>0, and for k>1:

D_k(n) = D_(k-1)(n-1) + D_k(n-k) if n >= 3*k-1, D_(k-1)(n-1) if 2*k-1 <= n <= 3*k-2, 0 if 1 <= n <= 2*k-2.

In the following, T(n,k) = D_k(n).