A002718 Number of bicoverings of an n-set. (Formerly M4559 N1941)

1, 0, 1, 8, 80, 1088, 19232, 424400, 11361786, 361058000, 13386003873, 570886397340, 27681861184474, 1511143062540976, 92091641176725504, 6219762391554815200, 462595509951068027741, 37676170944802047077248, 3343539821715571537772071, 321874499078487207168905840

Offset

0,4

Comments

Another description: number of proper 2-covers of [1,...,n].

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Peter Cameron, Thomas Prellberg, Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240 (see t_n).

L. Comtet, Birecouvrements et birevêtements d’un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]

Formula

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

Stirling_2 transform of A060053.

The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(e^x-1).

a(n) = Sum_{m=0..n + floor(n/2); k=0..n; s=0..min(m/2,k); t=0..m-2s} Stirling2(n,k) * k!/m! * binomial(m,2s) * A001147(s) * (-1)^(m+s+t) * binomial(m-2s,t) * binomial(t*(t-1)/2,k-s). Interpret m as the number of blocks in a bicovering, k the number of clumps of points that are always all together in blocks. This formula counts bicoverings by quotienting them to the clumpless case (an operation which preserves degree) and counting incidence matrices of those, and counts those matrices as the transposes of incidence matrices of labeled graphs with no isolated points and no isolated edges. - David Pasino, Jul 09 2016

Example

For n=3, there are 8 collections of distinct subsets of {1,2,3} with the property that each of 1, 2, and 3 appears in exactly two subsets:
{1,2,3},{1,2},{3}
{1,2,3},{1,3},{2}
{1,2,3},{2,3},{1}
{1,2,3},{1},{2},{3}
{1,2},{1,3},{2,3}
{1,2},{1,3},{2},{3}
{1,2},{2,3},{1},{3}
{1,3},{2,3},{1},{2}
Therefore a(3) = 8. - Michael B. Porter, Jul 16 2016

Mathematica

nmax = 16; imax = 2*(nmax - 2); egf := E^(-x - 1/2*x^2*(E^y - 1))*Sum[(x^i/i!)*E^(Binomial[i, 2]*y), {i, 0, imax}]; fx = CoefficientList[Series[egf, {y, 0, imax}], y]*Range[0, imax]!; a[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3] // Total; Table[ a[n] , {n, 2, nmax}] (* Jean-François Alcover, Apr 04 2013 *)

Crossrefs

Cf. A020554, A002719, A003462, A059945-A059951, A060053. Row sums of A059443.

Sequence in context: A234596 A060375 A097815 * A222825 A057707 A222671

Adjacent sequences: A002715 A002716 A002717 * A002719 A002720 A002721

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Feb 18 2001

a(0), a(1) prepended by Alois P. Heinz, Jul 29 2016

Status

approved