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A020554 Number of multigraphs on n labeled edges (without loops). 14
1, 1, 3, 16, 139, 1750, 29388, 624889, 16255738, 504717929, 18353177160, 769917601384, 36803030137203, 1984024379014193, 119571835094300406, 7995677265437541258, 589356399302126773920, 47609742627231823142029, 4193665147256300117666879 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Or, number of bicoverings of an n-set.

Or, number of 2-covers of [1,...,n].

Also the number of set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

REFERENCES

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

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 s_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.]

G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

FORMULA

E.g.f.: exp(-3/2+exp(x)/2)*Sum(exp(binomial(n, 2)*x)/n!, n=0..infinity) [Comtet]. - Vladeta Jovovic, Apr 27 2004

E.g.f. (an equivalent version in Maple format): G:=exp(-1+(exp(z)-1)/2)*sum(exp(s*(s-1)*z/2)/s!, s=0..infinity);

E.g.f.: exp((exp(x)-1)/2)*Sum(A020556(n)*(x/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004

Stirling_2 transform of A014500.

The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).

EXAMPLE

From Gus Wiseman, Jul 18 2018: (Start)

The a(3) = 16 set multipartitions of {1, 1, 2, 2, 3, 3}:

  (123)(123)

  (1)(23)(123) (2)(13)(123) (3)(12)(123) (12)(13)(23)

  (1)(1)(23)(23) (1)(2)(3)(123) (1)(2)(13)(23) (1)(3)(12)(23) (2)(2)(13)(13) (2)(3)(12)(13) (3)(3)(12)(12)

  (1)(1)(2)(3)(23) (1)(2)(2)(3)(13) (1)(2)(3)(3)(12)

  (1)(1)(2)(2)(3)(3)

(End)

MATHEMATICA

Ceiling[ CoefficientList[ Series[ Exp[ -1 + (Exp[ z ] - 1)/2 ]Sum[ Exp[ s(s - 1)z/2 ]/s!, {s, 0, 21} ], {z, 0, 9} ], z ] Table[ n!, {n, 0, 9} ] ] (* Mitch Harris, May 01 2004 *)

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

Table[Length[Select[mps[Ceiling[Range[1/2, n, 1/2]]], And@@UnsameQ@@@#&]], {n, 5}] (* Gus Wiseman, Jul 18 2018 *)

CROSSREFS

Row 2 of A188392.

Cf. A002718, A007716, A020555, A050535, A094574, A316974.

Sequence in context: A006057 A305472 A002719 * A062874 A321239 A109398

Adjacent sequences:  A020551 A020552 A020553 * A020555 A020556 A020557

KEYWORD

nonn,nice,easy

AUTHOR

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe

STATUS

approved

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Last modified June 22 06:25 EDT 2021. Contains 345372 sequences. (Running on oeis4.)