A060070 Number of T_0-tricoverings of an n-set.

1, 0, 0, 5, 175, 9426, 751365, 84012191, 12644839585, 2479642897109, 617049443550205, 190678639438170502, 71860665148118443795, 32527628234581386962713, 17454341903042193018433239, 10978059489008346809004564072, 8013452442154510131205645967978

Offset: 0

Author

Vladeta Jovovic, Feb 21 2001

Keywords

• nonn

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 = 0..100
• Vladeta Jovovic, T_0-tricoverings of a 4-set

Crossrefs

Row n=3 of A331039.

Row sums of A059530.

Cf. A060051, A060053, A060069, A060486.

Programs

PARI

seq(n)={my(m=2*n, y='y + O('y^(n+1))); Vec(serlaplace(subst(Pol(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!)), x, 1)))} \\ Andrew Howroyd, Jan 30 2020

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!.

a(n) = Sum_{k=0..n} Stirling1(n, k)*A060486(k). - Andrew Howroyd, Jan 08 2020

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 08 2020