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
KEYWORD
nonn,tabf
AUTHOR
Vladeta Jovovic, Feb 22 2001
STATUS
approved