A284592 Square array read by antidiagonals: T(n,k) is the number of pairs of partitions of n and k respectively, such that the pair of partitions have no part in common.

1, 1, 1, 2, 0, 2, 3, 1, 1, 3, 5, 1, 2, 1, 5, 7, 2, 3, 3, 2, 7, 11, 2, 5, 4, 5, 2, 11, 15, 4, 6, 7, 7, 6, 4, 15, 22, 4, 10, 8, 12, 8, 10, 4, 22, 30, 7, 12, 14, 14, 14, 14, 12, 7, 30, 42, 8, 18, 16, 24, 16, 24, 16, 18, 8, 42, 56, 12, 23, 25, 28, 28, 28, 28, 25, 23, 12, 56

Offset: 0

Author

Peter Bala, Mar 30 2017

Keywords

• nonn
• tabl
• easy

Comments

Compare with A284593.

Examples

			Square array begins
  n\k|  0  1  2  3  4  5   6   7   8   9  10
- - - - - - - - - - - - - - - - - - - - - - -
  0  |  1  1  2  3  5  7  11  15  22  30  42: A000041
  1  |  1  0  1  1  2  2   4   4   7   8  12: A002865
  2  |  2  1  2  3  5  6  10  12  18  23  32
  3  |  3  1  3  4  7  8  14  16  25  31  44
  4  |  5  2  5  7 12 14  24  28  43  54  76
  5  |  7  2  6  8 14 16  28  31  49  60  85
  6  | 11  4 10 14 24 28  48  55  85 106 149
  7  | 15  4 12 16 28 31  55  60  95 115 163
  8  | 22  7 18 25 43 49  85  95 148 182 256
  9  | 30  8 23 31 54 60 106 115 182 220 311
  10 | 42 12 32 44 76 85 149 163 256 311 438
  ...
T(4,3) = 7: the 7 pairs of partitions of 4 and 3 with no parts in common are (4, 3), (4, 2 + 1), (4, 1 + 1 + 1), (2 + 2, 3), (2 + 2, 1 + 1 + 1), (2 + 1 + 1 , 3) and (1 + 1 + 1 + 1, 3).
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened
• H. S. Wilf, Lectures on Integer Partitions

Crossrefs

Cf. A000041 (row 0), A002865 (row 1), A015128 (antidiagonal sums), A284593.

Main diagonal gives A054440 or 2*A260669 (for n>0).

Programs

Maple

#A284592 as a square array
ser := taylor(taylor(mul(1 + x^j/(1 - x^j) + y^j/(1 - y^j), j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, k, i) option remember; `if`(n=0 and
       (k=0 or i=1), 1, `if`(i<1, 0, b(n, k, i-1)+
       add(b(sort([n-i*j, k])[], i-1), j=1..n/i)+
       add(b(sort([n, k-i*j])[], i-1), j=1..k/i)))
    end:
A:= (n, k)-> (l-> b(l[1], l[2]$2))(sort([n, k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 02 2017

Mathematica

Table[Total@ Boole@ Map[! IntersectingQ @@ Map[Union, #] &, Tuples@ {IntegerPartitions@ #, IntegerPartitions@ k}] &[n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 02 2017 *)
b[n_, k_, i_] := b[n, k, i] = If[n == 0 &&
     (k == 0 || i == 1), 1, If[i < 1, 0, b[n, k, i - 1] +
     Sum[b[Sequence @@ Sort[{n - i*j, k}], i - 1], {j, 1, n/i}] +
     Sum[b[Sequence @@ Sort[{n, k - i*j}], i - 1], {j, 1, k/i}]]];
A[n_, k_] := Function [l, b[l[[1]], l[[2]], l[[2]]]][Sort[{n, k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

Formula

O.g.f. Product_{j >= 1} (1 + x^j/(1 - x^j) + y^j/(1 - y^j)) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A015128.