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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.

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