A035348 Triangle of a(n,k) = number of k-member minimal covers of an n-set (n >= k >= 1).
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
Offset: 1
Examples
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; ...
Links
- Alois P. Heinz, Rows n = 1..75, flattened
- R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
- T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
- A. J. Macula, Lewis Carroll and the enumeration of minimal covers, Math. Mag., 68 (1995), 269-274.
- Eric Weisstein's World of Mathematics, Minimal Cover
Programs
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Maple
a:= (n, k)-> add(binomial(2^k-k-1, m-k)*m! *Stirling2(n, m), m=k..min(n, 2^k-1))/k!: seq(seq(a(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jul 02 2013 -
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 *) -
PARI
{a(n, k) = sum(i=0, k, (-1)^i * binomial(k, i) * (2^k-1-i)^n) / k!} /* Michael Somos, Aug 05 1999 */
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
Extensions
Entry improved by Michael Somos
Explicit formulas added by N. J. A. Sloane, Aug 05 2011
Comments