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%I #30 Jun 10 2021 07:41:36
%S 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,
%T 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,
%U 24,16,24,16,18,8,42,56,12,23,25,28,28,28,28,25,23,12,56
%N 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.
%C Compare with A284593.
%H Alois P. Heinz, <a href="/A284592/b284592.txt">Antidiagonals n = 0..200, flattened</a>
%H H. S. Wilf, <a href="https://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf">Lectures on Integer Partitions</a>
%F 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).
%F Antidiagonal sums are A015128.
%e Square array begins
%e n\k| 0 1 2 3 4 5 6 7 8 9 10
%e - - - - - - - - - - - - - - - - - - - - - - -
%e 0 | 1 1 2 3 5 7 11 15 22 30 42: A000041
%e 1 | 1 0 1 1 2 2 4 4 7 8 12: A002865
%e 2 | 2 1 2 3 5 6 10 12 18 23 32
%e 3 | 3 1 3 4 7 8 14 16 25 31 44
%e 4 | 5 2 5 7 12 14 24 28 43 54 76
%e 5 | 7 2 6 8 14 16 28 31 49 60 85
%e 6 | 11 4 10 14 24 28 48 55 85 106 149
%e 7 | 15 4 12 16 28 31 55 60 95 115 163
%e 8 | 22 7 18 25 43 49 85 95 148 182 256
%e 9 | 30 8 23 31 54 60 106 115 182 220 311
%e 10 | 42 12 32 44 76 85 149 163 256 311 438
%e ...
%e 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).
%p #A284592 as a square array
%p ser := taylor(taylor(mul(1 + x^j/(1 - x^j) + y^j/(1 - y^j), j = 1..10), x, 11), y, 11):
%p convert(ser, polynom):
%p s := convert(%, polynom):
%p with(PolynomialTools):
%p for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
%p # second Maple program:
%p b:= proc(n, k, i) option remember; `if`(n=0 and
%p (k=0 or i=1), 1, `if`(i<1, 0, b(n, k, i-1)+
%p add(b(sort([n-i*j, k])[], i-1), j=1..n/i)+
%p add(b(sort([n, k-i*j])[], i-1), j=1..k/i)))
%p end:
%p A:= (n, k)-> (l-> b(l[1], l[2]$2))(sort([n, k])):
%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Apr 02 2017
%t 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 *)
%t b[n_, k_, i_] := b[n, k, i] = If[n == 0 &&
%t (k == 0 || i == 1), 1, If[i < 1, 0, b[n, k, i - 1] +
%t Sum[b[Sequence @@ Sort[{n - i*j, k}], i - 1], {j, 1, n/i}] +
%t Sum[b[Sequence @@ Sort[{n, k - i*j}], i - 1], {j, 1, k/i}]]];
%t A[n_, k_] := Function [l, b[l[[1]], l[[2]], l[[2]]]][Sort[{n, k}]];
%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 07 2021, after _Alois P. Heinz_ *)
%Y Cf. A000041 (row 0), A002865 (row 1), A015128 (antidiagonal sums), A284593.
%Y Main diagonal gives A054440 or 2*A260669 (for n>0).
%K nonn,tabl,easy
%O 0,4
%A _Peter Bala_, Mar 30 2017
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