A060053 -id:A060053 - cp's OEIS frontend

A002718 Number of bicoverings of an n-set.

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

Offset: 0

N. J. A. Sloane

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

			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

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).

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.]

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

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 *)

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

More terms from Vladeta Jovovic, Feb 18 2001

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

A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 1, 2, 9, 70, 794, 12055, 233238, 5556725, 158931613, 5350854707, 208746406117, 9315261027289, 470405726166241, 26636882237942128, 1678097862705130667, 116818375064650241036, 8932347052564257212796, 746244486452472386213939, 67796741482683128375533560

Offset: 0

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

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter Cameron, Thomas Prellberg, Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240 (see u_n).
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]

Row n=2 of A331126.

Cf. A020554, A020555, A014501, A060053.

read("transforms") ;
A020556 := proc(n) local k; add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end proc:
A014500 := proc(n) local i,gexp,lexp;
gexp := [seq(1/2^i/i!,i=0..n+1)] ;
lexp := add( A020556(i)*((log(1+x))/2)^i/i!,i=0..n+1) ;
lexp := taylor(lexp,x=0,n+1) ;
lexp := gfun[seriestolist](lexp,'ogf') ;
CONV(gexp,lexp) ; op(n+1,%)*n! ; end proc:
seq(A014500(n),n=0..20) ; # R. J. Mathar, Jul 03 2011

max = 20; A020556[n_] := Sum[(-1)^(n+k)*Binomial[n, k]*BellB[n+k], {k, 0, n}]; egf = Exp[x/2]*Sum[A020556[n]*(Log[1+x]/2)^n/n!, {n, 0, max}] + O[x]^max; CoefficientList[egf, x]*Range[0, max-1]! (* Jean-François Alcover, Feb 19 2017, after Vladeta Jovovic *)

\\ here egf1 is A020556 as e.g.f.
egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i,k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
seq(n)={my(B=egf1(n), L=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(x/2 + O(x*x^n))*sum(k=0, n, polcoef(B ,k)*L^k)))} \\ Andrew Howroyd, Jan 13 2020

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004
E.g.f.: exp(x/2)*Sum(A020556(n)*(log(1+x)/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004
Binomial transform of A060053.
The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).
The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).

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

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Feb 21 2001

nonn

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.

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

Andrew Howroyd, Table of n, a(n) for n = 0..100
Vladeta Jovovic, T_0-tricoverings of a 4-set

Row n=3 of A331039.
Row sums of A059530.
Cf. A060051, A060053, A060069, A060486.

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

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

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

A331039 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set-systems on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 43, 5, 0, 0, 1, 0, 1, 518, 175, 1, 0, 0, 1, 0, 1, 8186, 9426, 272, 0, 0, 0, 1, 0, 1, 163356, 751365, 64453, 205, 0, 0, 0, 1, 0, 1, 3988342, 84012191, 23553340, 248685, 80, 0, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 08 2020

An n-regular set-system is a finite set of nonempty sets in which each element appears in n blocks.
A set-system is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column and rows in decreasing lexicographic order.

			Array begins:
==========================================================
n\k | 0 1 2 3   4       5           6                7
----+-----------------------------------------------------
  0 | 1 1 0 0   0       0           0                0 ...
  1 | 1 1 1 1   1       1           1                1 ...
  2 | 1 0 1 5  43     518        8186           163356 ...
  3 | 1 0 0 5 175    9426      751365         84012191 ...
  4 | 1 0 0 1 272   64453    23553340      13241130441 ...
  5 | 1 0 0 0 205  248685   421934358    1176014951129 ...
  6 | 1 0 0 0  80  620548  5055634889   69754280936418 ...
  7 | 1 0 0 0  15 1057989 43402628681 2972156676325398 ...
  ...
The A(2,3) = 5 matrices are:
  [1 1 1]    [1 1 0]    [1 1 0]    [1 0 1]    [1 1 0]
  [1 0 0]    [1 0 1]    [1 0 0]    [1 0 0]    [1 0 1]
  [0 1 0]    [0 1 0]    [0 1 1]    [0 1 1]    [0 1 1]
  [0 0 1]    [0 0 1]    [0 0 1]    [0 1 0]
The corresponding set-systems are:
  {{1,2,3}, {1}, {2}, {3}},
  {{1,2}, {1,3}, {2,3}},
  {{1,2}, {1,3}, {2}, {3}},
  {{1,2}, {1}, {2,3}, {3}},
  {{1,3}, {1}, {2,3}, {2}}.

Andrew Howroyd, Table of n, a(n) for n = 0..209

Rows n=1..4 are A000012, A060053, A060070, A331655.

Cf. A188445, A330942, A330964, A331126, A331160, A331161, A331569, A331654.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}

A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188445(n, j) for n, k >= 1.
A(n, k) = 0 for k >= 1, n > 2^(k-1).
A331654(n) = Sum_{d|n} A(n/d, d).

A060090 Number of ordered bicoverings of an unlabeled n-set.

Original entry on oeis.org

1, 0, 3, 23, 290, 4298, 79143, 1702923, 42299820, 1188147639, 37276597020, 1291633545897, 48995506718702, 2019395409175529, 89864601931874318, 4294295828157319651, 219321170795303112118, 11922219151375200468886

Offset: 0

Vladeta Jovovic, Feb 25 2001

nonn

			There are 23 ordered bicoverings of an unlabeled 3-set, 7 3-block bicoverings:
1 ( { 3 }, { 1, 2 }, { 1, 2, 3 } )
2 ( { 3 }, { 1, 2, 3 }, { 1, 2 } )
3 ( { 2, 3 }, { 1 }, { 1, 2, 3 } )
4 ( { 2, 3 }, { 1, 3 }, { 1, 2 } )
5 ( { 2, 3 }, { 1, 2, 3 }, { 1 } )
6 ( { 1, 2, 3 }, { 3 }, { 1, 2 } )
7 ( { 1, 2, 3 }, { 2, 3 }, { 1 } )
and 16 4-block bicoverings:
1 ( { 3 }, { 2 }, { 1 }, { 1, 2, 3 } )
2 ( { 3 }, { 2 }, { 1, 3 }, { 1, 2 } )
3 ( { 3 }, { 2 }, { 1, 2 }, { 1, 3 } )
4 ( { 3 }, { 2 }, { 1, 2, 3 }, { 1 } )
5 ( { 3 }, { 2, 3 }, { 1 }, { 1, 2 } )
6 ( { 3 }, { 2, 3 }, { 1, 2 }, { 1 } )
7 ( { 3 }, { 1, 2 }, { 2 }, { 1, 3 } )
8 ( { 3 }, { 1, 2 }, { 2, 3 }, { 1 } )
9 ( { 3 }, { 1, 2, 3 }, { 2 }, { 1 } )
10 ( { 2, 3 }, { 3 }, { 1 }, { 1, 2 } )
11 ( { 2, 3 }, { 3 }, { 1, 2 }, { 1 } )
12 ( { 2, 3 }, { 1 }, { 3 }, { 1, 2 } )
13 ( { 2, 3 }, { 1 }, { 1, 3 }, { 2 } )
14 ( { 2, 3 }, { 1, 3 }, { 2 }, { 1 } )
15 ( { 2, 3 }, { 1, 3 }, { 1 }, { 2 } )
16 ( { 1, 2, 3 }, { 3 }, { 2 }, { 1 } )

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row n=2 of A331571.
Row sums of A060092.
Cf. A060069, A060070, A060051, A060052, A060053, A002718, A059443.

seq(n)={my(m=3*n\2, y='y + O('y^(n+1))); Vec(subst(Pol(serlaplace(exp(-x - x^2*y/(2*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 2)*x^k/k!))), x, 1))} \\ Andrew Howroyd, Jan 30 2020

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

A060051 Number of n-block r-bicoverings.

Original entry on oeis.org

1, 0, 0, 2, 79, 82117, 4936900199, 27555467226181396, 20554872166566046969648895, 2786548447182420815380482508924733911, 89607283195144164483079065133414172790220498449945, 864608448649084311874549352448884076627916391005243593208944730790

Offset: 0

Vladeta Jovovic, Feb 15 2001

A bicovering is an r-bicovering if the intersection of every two blocks contains at most one element.

			There are 2 3-block r-bicoverings: {{1},{2},{1,2}} and {{1,2},{1,3},{2,3}}.

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

Andrew Howroyd, Table of n, a(n) for n = 0..30

Column sums of A060052.

Cf. A002718, A060053, A060069, A060070.

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

Terms a(11) and beyond from Andrew Howroyd, Jan 30 2020

A060052 Triangle read by rows: T(n,k) gives number of r-bicoverings of an n-set with k blocks, n >= 2, k = 3..n+floor(n/2).

Original entry on oeis.org

1, 1, 4, 0, 15, 25, 3, 0, 30, 222, 226, 40, 0, 30, 1230, 3670, 2706, 535, 15, 0, 0, 5040, 39900, 69450, 40405, 8141, 420, 0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105, 0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040

Offset: 2

Vladeta Jovovic, Feb 15 2001

A bicovering is r-bicovering if intersection of every two blocks contains at most one element.

			Triangle starts:
[1],
[1, 4],
[0, 15, 25, 3],
[0, 30, 222, 226, 40],
[0, 30, 1230, 3670, 2706, 535, 15],
[0, 0, 5040, 39900, 69450, 40405, 8141, 420],
[0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105],
[0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040],
...

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

Andrew Howroyd, Table of n, a(n) for n = 2..1802 (rows n=2..50)
Table

Row sums are A060053.
Column sums are A060051.
Cf. A059443, A059530, A060092, A060487, A060492, A276640, A331039.

\\ returns k-th column as vector.
C(k)=if(k<3, [], Vecrev(serlaplace(polcoef(exp(-x-1/2*x^2*y + O(x*x^k))*sum(i=0, 3*k\2, (1+y)^binomial(i, 2)*x^i/i!), k))/y)) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=(3*n\2), y='y + O('y^(n+1))); my(g=exp(-x-1/2*x^2*y + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 2)*x^k/k!)); Mat([Col(serlaplace(p), -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=2, matsize(A)[1], print(A[n, 3..3*n\2])) } \\ Andrew Howroyd, Jan 30 2020

E.g.f.: A(x, y) = exp(-x-1/2*x^2*y)*Sum_{i>=0} (1+y)^binomial(i, 2)*x^i/i!.
T(n, k) = (n!/k!) * A276640(k, n). - David Pasino, Sep 22 2016
T(n,k) = 0 for n > binomial(k,2). - Andrew Howroyd, Jan 30 2020

Zeros inserted into data by Andrew Howroyd, Jan 30 2020

A060069 Number of n-block T_0-tricoverings.

Original entry on oeis.org

1, 0, 0, 0, 2, 82194, 9185157387760082, 5573096894405951375691132323893805593, 47933892393105239218152796441416602126447041437452022947424986090407628

Offset: 0

Vladeta Jovovic, Feb 19 2001

nonn

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.

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

Andrew Howroyd, Table of n, a(n) for n = 0..15

Column sums of A059530.

Cf. A060051, A060053, A060070, A060486.

E.g.f. for n-block T_0-tricoverings of a k-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!.

A060487 Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4).

Original entry on oeis.org

1, 3, 1, 7, 57, 95, 43, 3, 35, 717, 3107, 4520, 2465, 445, 12, 155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70, 651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

			Triangle begins:
  [1, 3, 1];
  [7, 57, 95, 43, 3];
  [35, 717, 3107, 4520, 2465, 445, 12];
  [155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70];
  [651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465];
   ...
There are 205 tricoverings of a 4-set(cf. A060486): 7 4-block, 57 5-block, 95 6-block, 43 7-block and 3 8-block tricoverings.

Andrew Howroyd, Table of n, a(n) for n = 3..1157

Columns include A060483, A060484, A060485.
Row sums are A060486.
Cf. A006095, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
row(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])*y^(m-n)/(1+y))}
for(n=3, 8, print(Vecrev(row(3,n)))); \\ Andrew Howroyd, Dec 23 2018

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

A060486 Tricoverings of an n-set.

Original entry on oeis.org

1, 0, 0, 5, 205, 11301, 904580, 101173251, 15207243828, 2975725761202, 738628553556470, 227636079973503479, 85554823285296622543, 38621481302086460057613, 20669385794052533823555309, 12966707189875262685801947906, 9441485712482676603570079314728

Offset: 0

Vladeta Jovovic, Mar 20 2001

nonn

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

			There are 1 4-block tricovering, 3 5-block tricoverings and 1 6-block tricovering of a 3-set (cf. A060487), so a(3)=5.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row 3 of A188445.

Cf. A006095, A060483-A060485, (row sums of) A060487, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

Terms a(11) and beyond from Andrew Howroyd, Dec 15 2018

cp's OEIS Frontend

A002718 Number of bicoverings of an n-set.

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A014500 Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.

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

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A331039 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set-systems on a k-set.

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A060090 Number of ordered bicoverings of an unlabeled n-set.

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A060051 Number of n-block r-bicoverings.

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A060052 Triangle read by rows: T(n,k) gives number of r-bicoverings of an n-set with k blocks, n >= 2, k = 3..n+floor(n/2).

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