|
|
A035348
|
|
Triangle of a(n,k) = number of k-member minimal covers of an n-set (n >= k >= 1).
|
|
13
|
|
|
1, 1, 1, 1, 6, 1, 1, 25, 22, 1, 1, 90, 305, 65, 1, 1, 301, 3410, 2540, 171, 1, 1, 966, 33621, 77350, 17066, 420, 1, 1, 3025, 305382, 2022951, 1298346, 100814, 988, 1, 1, 9330, 2619625, 47708115, 83384427, 18151560, 549102, 2259, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
COMMENTS
|
These are what Clarke calls "Minimal disordered k-covers of labeled n-set".
|
|
LINKS
|
|
|
FORMULA
|
a(n,k) = Sum_{j >= 0} (-1)^j * binomial(k,j) * (2^k-1-j)^n. [Hearne-Wagner]
a(n,k) = (1/k!) * Sum_{j >= k} binomial(2^k-k-1,j-k)*j!*Stirling2(n,j). [Macula]
E.g.f.: Sum_{n>=0} (exp(y)-1)^n*exp(y*(2^n-n-1))*x^n/n!. - Vladeta Jovovic, May 08 2004
|
|
EXAMPLE
|
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 25, 22, 1;
1, 90, 305, 65, 1,
1, 301, 3410, 2540, 171, 1;
1, 966, 33621, 77350, 17066, 420, 1;
1, 3025, 305382, 2022951, 1298346, 100814, 988, 1;
...
|
|
MAPLE
|
a:= (n, k)-> add(binomial(2^k-k-1, m-k)*m!
*Stirling2(n, m), m=k..min(n, 2^k-1))/k!:
|
|
MATHEMATICA
|
a[n_, k_] := Sum[ (-1)^i*(2^k-i-1)^n / (i!*(k-i)!), {i, 0, k}]; Flatten[ Table[ a[n, k], {n, 1, 9}, {k, 1, n}]] (* Jean-François Alcover, Dec 13 2011, after PARI *)
|
|
PROG
|
(PARI) {a(n, k) = sum(i=0, k, (-1)^i * binomial(k, i) * (2^k-1-i)^n) / k!} /* Michael Somos, Aug 05 1999 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|