A282575 Triangular array read by rows. T(n,k) is the number of minimal covers of an n-set with exactly k points that are in more than one set of the cover, n>=0, 0<=k<=max(0,n-2).

1, 1, 2, 5, 3, 15, 28, 6, 52, 210, 190, 10, 203, 1506, 3360, 1340, 15, 877, 10871, 48321, 60270, 9065, 21, 4140, 80592, 636300, 1820056, 1132880, 57512, 28, 21147, 618939, 8081928, 45455676, 76834926, 21067452, 344316, 36, 115975, 4942070, 101684115, 1027544400, 3860929170, 3406410252, 377190240, 1966440, 45

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..100, flattened

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

Formula

E.g.f.: (exp(x) - 1)^n/n!*exp(y*(2^n - n - 1)*x).

Example

Triangle T(n,k) begins:
:    1;
:    1;
:    2;
:    5,     3;
:   15,    28,      6;
:   52,   210,    190,      10;
:  203,  1506,   3360,    1340,      15;
:  877, 10871,  48321,   60270,    9065,    21;
: 4140, 80592, 636300, 1820056, 1132880, 57512, 28;

Maple

T:= (n, k)-> binomial(n, k)*add(Stirling2(n-k, j)*(2^j-j-1)^k, j=0..n-k):
seq(seq(T(n, k), k=0..max(0, n-2)), n=0..12);  # Alois P. Heinz, Feb 18 2017

Mathematica

nn = 8; Drop[Map[Select[#, # > 0 &] &, Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! Exp[y (2^n - n - 1) x], {n, 0, nn}], {x, 0, nn}], {x, y}]], 1] // Grid

Crossrefs

Cf. A035348. Row sums A046165. Column k=0 A000110. Column k=1 A003466.

Mirrored triangle gives A035347.

Sequence in context: A243066 A368316 A181921 * A002565 A063703 A109619

Adjacent sequences: A282572 A282573 A282574 * A282576 A282577 A282578

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 18 2017

Status

approved