A282575 -id:A282575 - cp's OEIS frontend

A046165 Number of minimal covers of n objects.

1, 1, 2, 8, 49, 462, 6424, 129425, 3731508, 152424420, 8780782707, 710389021036, 80610570275140, 12815915627480695, 2855758994821922882, 892194474524889501292, 391202163933291014701953, 240943718535427829240708786, 208683398342300491409959279244

Offset: 0

Eric W. Weisstein

nonn

No edge of a minimal cover can be a subset of any other, so minimal covers are antichains, but the converse is not true. - Gus Wiseman, Jul 03 2019

a(n) is the number of undirected graphs on n nodes for which the intersection number and independence number are equal. See Proposition 2.3.7 and Theorem 2.3.3 of the Deligeorgaki et al. paper below. - Alex Markham, Oct 13 2022

			From _Gus Wiseman_, Jul 02 2019: (Start)
The a(1) = 1 through a(3) = 8 minimal covers:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1,2},{1,3}}
                    {{1,2},{2,3}}
                    {{1},{2},{3}}
                    {{1,3},{2,3}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..113
Damian Bursztyn, François Goasdoué, and Ioana Manolescu, Optimizing Reformulation-based Query Answering in RDF, [Research Report] RR-8646, INRIA Saclay. 2014.
D. Deligeorgaki, A. Markham, P. Misra, and L. Solus, Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks, arXiv:2210.00822 [stat.ME], 2022.
Giovanni Resta, Illustration of a(4)=49.
Eric Weisstein's World of Mathematics, Minimal Cover

Cf. A035348, A000371, A003465.
Antichain covers are A006126.
Minimal covering simple graphs are A053530.
Maximal antichains are A326358.
Row sums of A035347 or of A282575.
Cf. A000372, A003182, A006602, A261005, A305844, A307249, A326359.

a:= n-> add(add((-1)^i* binomial(k,i) *(2^k-1-i)^n, i=0..k)/k!, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2008

Table[Sum[Sum[Binomial[n,i]StirlingS2[i,k](2^k-k-1)^(n-i),{i,k,n}],{k,2,n}]+1,{n,1,20}] (* Geoffrey Critzer, Jun 27 2013 *)

E.g.f.: Sum_{n>=0} (exp(x)-1)^n*exp(x*(2^n-n-1))/n!. - Vladeta Jovovic, May 08 2004
a(n) = Sum_{k=1..n} Sum_{i=k..n} C(n,i)*Stirling2(i,k)*(2^k - k - 1)^(n - i). - Geoffrey Critzer, Jun 27 2013
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

a(0)=1 prepended by Alois P. Heinz, Feb 18 2017

A003466 Number of minimal covers of an n-set that have exactly one point which appears in more than one set in the cover.

Original entry on oeis.org

0, 3, 28, 210, 1506, 10871, 80592, 618939, 4942070, 41076508, 355372524, 3198027157, 29905143464, 290243182755, 2920041395248, 30414515081650, 327567816748638, 3643600859114439, 41809197852127240, 494367554679088923, 6017481714095327410

Offset: 2

N. J. A. Sloane

nonn

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

Robert Israel, Table of n, a(n) for n = 2..510
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.

Cf. A046165.

Column k=1 of A282575.

seq(n*add((2^k-k-1)*Stirling2(n-1,k),k=1..n-1), n = 2 .. 30); # Robert Israel, May 21 2015

nn = 20; Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! (2^n - n - 1) x, {n, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2017 *)
a[2]=0;a[3]=3;a[4]=28;a[n_]:=n*Sum[(2^k-k-1)* StirlingS2[n-1,k], {k,1,n-1}];Table[a[n],{n,2,22}] (* Indranil Ghosh, Feb 20 2017 *)

a(n) = n * Sum_{k=1..n-1} (2^k-k-1) * S2(n-1,k) where S2(n,k) are the Stirling numbers of the second kind. - Sean A. Irvine, May 20 2015

a(n) = n * (A001861(n-1) - A005493(n-2) - A000110(n-1)). - Robert Israel, May 21 2015

More terms from Sean A. Irvine, May 20 2015

Title clarified by Geoffrey Critzer, Feb 18 2017

cp's OEIS Frontend

A046165 Number of minimal covers of n objects.

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