A060051 -id:A060051 - cp's OEIS frontend

A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

1, 0, 1, 5, 43, 518, 8186, 163356, 3988342, 116396952, 3985947805, 157783127673, 7131072006829, 364166073164914, 20827961078794845, 1323968417981743817, 92917890994442697487, 7157607311779373890120, 602043767970637640566684

Offset: 0

Vladeta Jovovic, Feb 15 2001

A bicovering is called an r-bicovering if the intersection of every two blocks contains at most one element.
Another name for this sequence is the number of restricted proper 2-covers of [1,...,n].
Number of T_0 2-regular set-systems on an n-set. - Andrew Howroyd, Jan 08 2020

			There are 5 r-bicoverings of a 3-set: 1 3-block bicovering {{1, 2}, {1, 3}, {2, 3}} and 4 4-block bicoverings {{1}, {2}, {3}, {1, 2, 3}}, {{2}, {3}, {1, 2}, {1, 3}}, {{1}, {3}, {1, 2}, {2, 3}}, {{1}, {2}, {1, 3}, {2, 3}}.
G.f. = 1 + x^2 + 5*x^3 + 43*x^4 + 518*x^5 + 8186*x^6 + 163356*x^7 + ...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. (See p. 203.)

Robert Gerbicz, Table of n, a(n) for n = 0..100
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, arXiv:0707.0664 [math.CO], 2007.
Peter Cameron, Thomas Prellberg, and Dudley Stark, Asymptotic enumeration of 2-covers and line graphs, Discrete Math. 310 (2010), no. 2, 230-240. (See v_n.)

Row 2 of A331039.
Row sums of A060052.
Cf. A060051, A002718, A059443, A003462, A059945-A059951.

A060053 := proc(n) local h, m; h := (m,n) -> add((-1/2)^k*binomial(m*(m-1)/2,n-k)/k!, k=0..n); n!*add(h(m,n)/m!, m=0..3*n); ceil(evalf(%/exp(1),99)) end: seq(A060053(i), i=0..18);
# Caveat computator! Limited accuracy. Do not use it for n > 50. - Peter Luschny, Jul 06 2011

f[n_] := FullSimplify[(n!/E)*Sum[(1/m!)*Sum[(-1/2)^k*Binomial[m*(m - 1)/2,
n - k]/k!, {k, 0, n}], {m, 0, Infinity}]] (* Robert G. Wilson v, Jul 03 2011 *)

a(n)=round(n!/exp(1)*sum(m=0,3*n+1,1/m!*sum(k=0,n,(-1/2)^k*binomial(m*(m-1)/2,n-k)/k!)))

\\ 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(A=egf1(n), B=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(-x/2 + O(x*x^n))*sum(k=0, n, polcoef(A,k)*B^k)))} \\ Andrew Howroyd, Jan 13 2020

E.g.f. for number of k-block r-bicoverings of an n-set is exp(-x-x^2*y/2)*Sum_{i=0..inf} (1+y)^binomial(i, 2)*x^i/i!.
a(n) = row sums of A060052.
Inverse binomial transform of A014500. - Vladeta Jovovic, Aug 22 2006
The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(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).
E.g.f.: exp(-x/2)*(Sum_{k>=0} A020556(k)*(log(1 + x)/2)^k/k!). - Andrew Howroyd, Jan 13 2020

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

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!.

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

A060483 Number of 5-block tricoverings of an n-set.

Original entry on oeis.org

3, 57, 717, 7845, 81333, 825237, 8300757, 83202645, 832809813, 8331237717, 83324947797, 833299785045, 8333199127893, 83332796486997, 833331185898837, 8333324743497045, 83333298973791573, 833333195894773077, 8333332783578305877, 83333331134311650645

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.

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (17,-84,148,-80).

Column k=5 of A060487.

Cf. A006095, A060484, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Vec(3*(1+2*x)/(x-1)/(2*x-1)/(4*x-1)/(10*x-1)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2013

a(n) = (1/5!)*(10^n - 15*4^n + 45*2^n - 40).
Generally, 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).
G.f.: 3*x^3*(2*x+1) / ((x-1)*(2*x-1)*(4*x-1)*(10*x-1)). - Colin Barker, Jan 11 2013

More terms from Colin Barker, Jan 11 2013

A060491 Number of ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

1, 0, 0, 184, 17488, 2780752, 689187720, 236477490418, 107317805999204, 62318195302890305, 45081693413563797127, 39762626850034005271588, 42009504510315968282400843, 52381340312720286113688037624, 76118747309505733406576769607755

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 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060492).

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

Row n=3 of A331571.
Row sums of A060492.
Cf. A060486, A060487, A060090, A060092, A060069, A060070, A060051, A060052, A060053, A002718, A059443.

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

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

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

A060484 Number of 6-block tricoverings of an n-set.

Original entry on oeis.org

1, 95, 3107, 75835, 1653771, 34384875, 700030507, 14116715435, 283432939691, 5679127043755, 113683003777707, 2274630646577835, 45502044971338411, 910133025632152235, 18203564201836161707, 364080180268471397035

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.

Andrew Howroyd, Table of n, a(n) for n = 3..200
Index entries for linear recurrences with constant coefficients, signature (45,-720,5220,-17664,25920,-12800).

Column k=6 of A060487.

Cf. A006095, A060483, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

With[{c=1/6!},Table[c(20^n-6*10^n-15*8^n+135*4^n-310*2^n+240),{n,3,20}]] (* or *) LinearRecurrence[{45,-720,5220,-17664,25920,-12800},{1,95,3107,75835,1653771,34384875},20] (* Harvey P. Dale, Jan 05 2017 *)

a(n) = (1/6!)*(20^n - 6*10^n - 15*8^n + 135*4^n - 310*2^n + 240) \\ Andrew Howroyd, Dec 15 2018

a(n) = (1/6!)*(20^n - 6*10^n - 15*8^n + 135*4^n - 310*2^n + 240).
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).
G.f.: -x^3*(800*x^3+448*x^2-50*x-1) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(10*x-1)*(20*x-1)). - Colin Barker, Jan 12 2013
a(n) = 45*a(n-1)-720*a(n-2)+5220*a(n-3)-17664*a(n-4)+25920*a(n-5)-12800*a(n-6). - Wesley Ivan Hurt, Oct 18 2021

cp's OEIS Frontend

A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

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

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

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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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A060069 Number of n-block T_0-tricoverings.

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

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A060486 Tricoverings of an n-set.

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