A035347 Triangle of a(n,k) = number of minimal covers of an n-set that cover k points of that set uniquely (n >= 1, k >= 1).

1, 0, 2, 0, 3, 5, 0, 6, 28, 15, 0, 10, 190, 210, 52, 0, 15, 1340, 3360, 1506, 203, 0, 21, 9065, 60270, 48321, 10871, 877, 0, 28, 57512, 1132880, 1820056, 636300, 80592, 4140, 0, 36, 344316, 21067452, 76834926, 45455676, 8081928, 618939, 21147, 0, 45

Offset

1,3

Links

Table of n, a(n) for n=1..47.

T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.

Formula

a(n, k) = C(n, k)*Sum_{j=1..k} S(k, j)*(2^j-j-1)^(n-k), where S(k, j) are Stirling numbers of the second kind.

E.g.f.: Sum_{k>=1} (exp(y*x) - 1)^k/k! * exp((2^k-k-1)x). - Geoffrey Critzer, Jun 28 2013

Example

1; 0,2; 0,3,5; 0,6,28,15; ...

Mathematica

a[n_, k_] := Binomial[n, k] * Sum[ StirlingS2[k, j]*(2^j - j - 1)^(n - k), {j, 1, k}]; a[n_, n_] := Sum[ StirlingS2[n, j], {j, 1, n}]; Flatten[ Table[a[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Jun 26 2012, from formula *)

Crossrefs

Cf. A056885 for unlabeled case. Row sums give A046165.

Sequence in context: A336973 A369017 A352846 * A094126 A293269 A038072

Adjacent sequences: A035344 A035345 A035346 * A035348 A035349 A035350

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Sep 06 2000

Status

approved