A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 2, 0, 1, 0, 4, 1, 1, 3, 1, 1, 0, 5, 2, 2, 2, 3, 0, 1, 0, 6, 2, 4, 2, 3, 3, 1, 1, 0, 8, 3, 4, 4, 3, 2, 5, 0, 1, 0, 10, 3, 5, 4, 7, 4, 3, 4, 1, 1, 0, 12, 5, 6, 6, 7, 7, 4, 3, 5, 0, 1, 0, 15, 5, 9, 7, 8, 6, 12, 3, 4, 6, 1, 1, 0

Offset: 0

Author

Gus Wiseman, Aug 17 2023

Keywords

• nonn
• tabl

Comments

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.

Examples

			Array begins:
  1  1  1  2  2  3  4   5   6   8   10   12  15   18   22   27
  0  1  0  1  1  1  2   2   3   3   5    5   7    8    10   12
  0  1  1  2  1  2  4   4   5   6   9    10  13   15   19   23
  0  1  0  3  2  2  4   4   6   7   11   11  15   17   22   27
  0  1  1  3  3  3  7   7   8   10  16   17  23   27   33   42
  0  1  0  3  2  4  7   6   9   9   17   17  23   26   33   43
  0  1  1  5  3  4  12  10  13  16  26   27  36   42   52   68
  0  1  0  4  3  3  10  11  13  13  27   25  35   40   51   67
  0  1  1  5  4  5  15  13  19  20  36   37  51   58   72   97
  0  1  0  6  4  5  14  13  18  23  42   39  54   61   78   105
  0  1  1  6  4  6  20  17  23  25  54   50  69   80   98   138
  0  1  0  6  4  5  19  16  23  24  54   55  71   80   103  144
  0  1  1  8  6  7  27  23  30  35  72   70  103  113  139  199
  0  1  0  7  5  6  24  21  29  31  75   68  95   115  139  201
  0  1  1  8  5  7  31  27  36  39  90   86  122  137  178  255
  0  1  0  9  6  8  31  27  38  42  100  93  129  148  187  289
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   3  1  2  0  1  0
   4  1  1  3  1  1  0
   5  2  2  2  3  0  1  0
   6  2  4  2  3  3  1  1  0
   8  3  4  4  3  2  5  0  1  0
  10  3  5  4  7  4  3  4  1  1  0
  12  5  6  6  7  7  4  3  5  0  1  0
  15  5  9  7  8  6 12  3  4  6  1  1  0
  18  7 10 11 10  9 10 10  5  4  6  0  1  0
  22  8 13 11 16  9 13 11 15  5  4  6  1  1  0
  27 10 15 15 17 17 16 13 13 14  6  4  8  0  1  0
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened

Crossrefs

Same as A116861 with offset 0 and rows reversed, non-strict version A364912.

Row n = 0 is A000009.

Row n = 1 is A096765.

Row n = 2 is A365005.

Column k = 0 is A000007.

Column k = 1 is A000012.

Column k = 2 is A000035.

Column k = 3 is A137719.

The main diagonal is A364910.

Left half has row sums A365002.

For not just strict partitions we have A365004, diagonal A364907.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A066328 adds up distinct prime indices.

A364350 counts combination-free strict partitions, complement A364839.

Cf. A007865, A085489, A108917, A237113, A323092, A364272, A364533, A364670, A364911, A364913.

Programs

Mathematica

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
t[n_,k_]:=Length[Join@@Table[combs[n,ptn],{ptn,Select[IntegerPartitions[k],UnsameQ@@#&]}]];
Table[t[k,n-k],{n,0,15},{k,0,n}]