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
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
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);