A284592 - OEIS

%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