A370005 Number T(n,k) of ordered 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, 2, 1, 1, 4, 3, 1, 1, 12, 7, 4, 1, 1, 16, 19, 8, 4, 1, 1, 48, 35, 23, 9, 4, 1, 1, 60, 83, 43, 24, 9, 4, 1, 1, 148, 143, 106, 47, 25, 9, 4, 1, 1, 220, 291, 186, 115, 48, 25, 9, 4, 1, 1, 438, 511, 397, 210, 119, 49, 25, 9, 4, 1, 1, 618, 949, 697, 444, 219, 120, 49, 25, 9, 4, 1, 1
Offset: 0
Examples
T(4,0) = 12: (1111,22), (1111,4), (211,4), (22,1111), (22,31), (22,4), (31,22), (31,4), (4,1111), (4,211), (4,22), (4,31).
T(4,1) = 7: (1111,31), (211,22), (211,31), (22,211), (31,1111), (31,211), (4,4).
T(4,2) = 4: (1111,211), (211,1111), (22,22), (31,31).
T(4,3) = 1: (211,211).
T(4,4) = 1: (1111,1111).
Triangle T(n,k) begins:
1;
0, 1;
2, 1, 1;
4, 3, 1, 1;
12, 7, 4, 1, 1;
16, 19, 8, 4, 1, 1;
48, 35, 23, 9, 4, 1, 1;
60, 83, 43, 24, 9, 4, 1, 1;
148, 143, 106, 47, 25, 9, 4, 1, 1;
220, 291, 186, 115, 48, 25, 9, 4, 1, 1;
438, 511, 397, 210, 119, 49, 25, 9, 4, 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: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$3)): seq(T(n), n=0..12);