A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 39, 89, 43, 3, 0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12, 0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70, 0, 0, 0, 0, 0, 5040, 536760, 6052730, 20660055, 29432319, 19826737, 6481160, 964495, 52430

Offset

3,6

Comments

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering. A covering of a set is a T_0-covering if for every two distinct elements of the set there exists a block of the covering containing one but not the other element.

References

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Links

Andrew Howroyd, Table of n, a(n) for n = 3..1674

Vladeta Jovovic, T_0-tricoverings of a 4-set

Formula

E.g.f. for k-block T_0-tricoverings of an n-set is exp(-x+1/2*x^2+1/3*x^3*y)*Sum_{i=0..inf}(1+y)^binomial(i, 3)*exp(-1/2*x^2*(1+y)^i)*x^i/i!.

T(n,k) = 0 for n > binomial(k, 3). - Andrew Howroyd, Jan 30 2020

Example

Triangle begins:
  [0, 0, 0, 0, 1, 3, 1],
  [0, 0, 0, 0, 1, 39, 89, 43, 3],
  [0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12],
  [0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70],
   ...
There are 5 = 1 + 3 + 1 T_0-tricoverings of a 3-set and 175 = 1 + 39 + 89 + 43 + 3 T_0-tricoverings of a 4-set, cf. A060070.

Prog

(PARI) \\ gets k-th column as vector
C(k)=if(k<4, [], Vecrev(serlaplace(polcoef(exp(-x + x^2/2 + x^3*y/3 + O(x*x^k))*sum(i=0, 2*k, (1+y)^binomial(i, 3)*exp(-x^2*(1+y)^i/2 + O(x*x^k))*x^i/i!), k))/y)) \\ Andrew Howroyd, Jan 30 2020
(PARI)
T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=exp(-x + x^2/2 + x^3*y/3 + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 3)*exp(-x^2*(1+y)^k/2 + O(x*x^m))*x^k/k!)); Mat([Col(serlaplace(p), -n) | p<-Vec(g)[2..m+1]]); }
{ my(A=T(8)); for(n=3, matsize(A)[1], print(concat([0], A[n, 1..2*n]))) } \\ Andrew Howroyd, Jan 30 2020

Crossrefs

Column sums are A060069.

Row sums are A060070.

Cf. A059443, A060052, A060092, A060487, A060492, A331039.

Sequence in context: A078529 A180017 A243827 * A193525 A049828 A342557

Adjacent sequences: A059527 A059528 A059529 * A059531 A059532 A059533

Keyword

nonn,tabf

Author

Vladeta Jovovic, Feb 22 2001

Status

approved