A370207 Number T(n,k) of unordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 6, 4, 3, 1, 1, 8, 10, 5, 3, 1, 1, 24, 18, 13, 6, 3, 1, 1, 30, 42, 23, 14, 6, 3, 1, 1, 74, 72, 55, 26, 15, 6, 3, 1, 1, 110, 146, 95, 61, 27, 15, 6, 3, 1, 1, 219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1, 309, 475, 351, 227, 115, 65, 28, 15, 6, 3, 1, 1

Offset: 0

Author

Alois P. Heinz, Feb 11 2024

Keywords

• nonn
• tabl

Examples

			T(4,0) = 6: (1111,22), (1111,4), (211,4), (22,31), (22,4), (31,4).
T(4,1) = 4: (1111,31), (211,22), (211,31), (4,4).
T(4,2) = 3: (1111,211), (22,22), (31,31).
T(4,3) = 1: (211,211).
T(4,4) = 1: (1111,1111).
Triangle T(n,k) begins:
    1;
    0,   1;
    1,   1,   1;
    2,   2,   1,   1;
    6,   4,   3,   1,  1;
    8,  10,   5,   3,  1,  1;
   24,  18,  13,   6,  3,  1,  1;
   30,  42,  23,  14,  6,  3,  1, 1;
   74,  72,  55,  26, 15,  6,  3, 1, 1;
  110, 146,  95,  61, 27, 15,  6, 3, 1, 1;
  219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1;
  ...
		

Links

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

Crossrefs

Column k=0 gives A260669.

Row sums and T(2n,n) give A086737.

Cf. A000041, A000217, A008284, A072233, A370005.

Programs

Maple

b:= proc(n, m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
      add(add(expand(b(sort([n-i*j, m-i*h])[], i-1)*
       x^min(j, h)), h=0..m/i), j=0..n/i)))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(g(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= (n, k)-> (coeff(b(n$3), x, k)+coeff(g(n$2), x, k))/2:
seq(seq(T(n, k), k=0..n), n=0..12);

Formula

T(n,k) = (A370005(n,k) + A072233(n,k))/2.