A005785 -id:A005785 - cp's OEIS frontend

A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5, 1, 1, 12, 51, 65, 28, 6, 1, 1, 16, 103, 230, 156, 43, 7, 1, 1, 20, 196, 736, 863, 336, 62, 8, 1, 1, 25, 348, 2197, 4571, 2864, 664, 86, 9, 1, 1, 30, 590, 6093, 22952, 25326, 8609, 1229, 115, 10, 1, 1, 36, 960

Offset: 1

Vladeta Jovovic, Jun 13 2000

Also number of unlabeled split graphs on n vertices and with a k-element clique (cf. A048194).

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   9,   4,   1;
  1,  9,  23,  17,   5,   1;
  1, 12,  51,  65,  28,   6,  1;
  1, 16, 103, 230, 156,  43,  7, 1;
  1, 20, 196, 736, 863, 336, 62, 8, 1;
  ...
There are four minimal covers of an unlabeled 3-set: one 1-cover {{1,2,3}}, two 2-covers {{1,2},{3}}, {{1,2},{1,3}} and one 3-cover {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
G. F. Royle, Counting Set Covers and Split Graphs, J. Integer Seqs., 3 (2000), #00.2.6.
Eric Weisstein's World of Mathematics, Minimal covers

Row sums give A048194.
Columns are A000012, A087811, A005783, A005784, A005785 A005786, A055066.
Diagonals are A000012, A000027, A005744, A005745, A005746, A005771, A005747, A005748.
Cf. A035348 for labeled case.
Cf. A002620, A028657.

\\ Needs A(n,m) from A028657.
T(n,k) = A(n-k, k) - if(kAndrew Howroyd, Feb 28 2023

T(n,k) = A028657(n,k) - A028657(n-1,k). - Andrew Howroyd, Feb 28 2023

A005783 Number of 3-covers of an unlabeled n-set.

Original entry on oeis.org

1, 3, 9, 23, 51, 103, 196, 348, 590, 960, 1506, 2290, 3393, 4905, 6945, 9651, 13185, 17739, 23542, 30846, 39954, 51206, 64986, 81730, 101935, 126141, 154967, 189093, 229269, 276325, 331182, 394830, 468372, 553002, 650016, 760824, 886963

Offset: 0

N. J. A. Sloane

Equals first differences of A002727. - Vladeta Jovovic, May 24 2000

Number of 3 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

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

Stefano Spezia, Table of n, a(n) for n = 0..10000 (terms for n = 1..1000 from T. D. Noe)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Masaaki Harada, Ken Saito, Binary linear complementary dual codes, arXiv:1802.06985 [math.CO], 2018.
Vladeta Jovovic, Binary matrices up to row and column permutations
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,-1,3,6,-6,-3,1,3,1,-3,1).

Column 3 of A055080.

Cf. A002727, A002620, A005784, A005785, A005786, A055066.

CoefficientList[Series[(x^6+x^4+2x^3+x^2+1)/((1-x^3)^2(1-x^2)^2 (1-x)^3),{x,0,50}],x] (* Harvey P. Dale, May 19 2011 *)

Vec(G(3, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^6+x^4+2*x^3+x^2+1)/((1-x^3)^2*(1-x^2)^2*(1-x)^3).
a(n) ~ n^6/4320. - Stefano Spezia, Aug 08 2022
a(n) = n^6/4320 + 7*n^5/1440 + 79*n^4/1728 + 35*n^3/144 + 2939*n^2/4320 + 8863*n/8640 + 1 + (n/16 + 7/32)*floor(n/2) + (n/9 + 11/27)*floor(n/3) + floor((n+1)/3)/27. - Vaclav Kotesovec, Aug 09 2022

More terms from Vladeta Jovovic, May 24 2000

a(0) = 1 prepended by Stefano Spezia, Aug 09 2022

A057968 Triangle T(n,k) of numbers of minimal 5-covers of an unlabeled n+5-set that cover k points of that set uniquely (k=5,..,n+5).

Original entry on oeis.org

1, 4, 1, 19, 7, 2, 91, 46, 16, 3, 436, 279, 115, 28, 5, 1991, 1563, 740, 221, 49, 7, 8651, 7978, 4309, 1524, 405, 75, 10, 35354, 37290, 22604, 9272, 2875, 659, 115, 13, 135617, 159948, 107584, 50058, 17840, 4866, 1042, 163, 18, 488312, 633211

Offset: 0

Vladeta Jovovic, Oct 17 2000

Row sums give A005785.

			[1], [4, 1], [19, 7, 2], [91, 46, 16, 3], [436, 279, 115, 28, 5], ...; there are 46 minimal 5-covers of an unlabeled 8-set that cover 6 points of that set uniquely.

More information

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965, A057966(labeled case), A057967.

T(n, k)=b(n, k)-b(n-1, k); b(n, k)=coefficient of x^k in (x^5/5!)*(Z(S_n; 27+5*x, 27+5*x^2, ...)+10*Z(S_n; 13+3*x, 27+5*x^2, 13+3*x^3, 27+5*x^4, ...)+15*Z(S_n; 7+x, 27+5*x^2, 7+x^3, 27+5*x^4, ...)+20*Z(S_n; 6+2*x, 6+2*x^2, 27+5*x^3, 6+2*x^4, 6+2*x^5, 27+5*x^6, ...)+20*Z(S_n; 4, 6+2*x^2, 13+3*x^3, 6+2*x^4, 4, 27+5*x^6, 4, 6+2*x^8, 13+3*x^9, 6+2*x^10, 4, 27+5*x^12, ...)+30*Z(S_n; 3+x, 7+x^2, 3+x^3, 27+5*x^4, 3+x^5, 7+x^6, 3+x^7, 27+5*x^8, ...)+24*Z(S_n; 2, 2, 2, 2, 27+5*x^5, 2, 2, 2, 2, 27+5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

A055066 Number of 7-covers of an unlabeled n-set.

Original entry on oeis.org

1, 7, 62, 664, 8609, 127415, 2004975, 31500927, 474504448, 6708348262, 88249739792, 1078567590128, 12269901302433, 130370516668917, 1298891291366245, 12182760243381355, 107979270564656625, 907568508195185203, 7256984238345563764, 55365443728411530716, 404091280028746802188

Offset: 0

Vladeta Jovovic, Jun 12 2000

nonn

Number of 7 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Binary matrices up to row and column permutations

Column 7 of A055080.

Cf. A005748, A002620, A005783, A005784, A005785, A005786.

Vec(G(7, x)*(1-x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

a(0)=1 prepended by Alois P. Heinz, Aug 08 2022

Terms a(17) and beyond from Andrew Howroyd, Feb 28 2023

A005784 Number of 4-covers of an unlabeled n-set.

Original entry on oeis.org

1, 4, 17, 65, 230, 736, 2197, 6093, 15864, 38960, 90837, 202005, 430577, 883057, 1748909, 3355213, 6252575, 11345602, 20089514, 34778306, 58964020, 98053576, 160151566, 257229974, 406739271, 633795181, 974126408, 1477999320, 2215409037, 3282874359, 4812278064

Offset: 0

N. J. A. Sloane

Number of 4 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

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

Stefano Spezia, Table of n, a(n) for n = 0..10000
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
Index entries for linear recurrences with constant coefficients, signature (5, -7, -1, 5, -1, 21, -33, 0, -4, 8, 68, -57, -3, -57, 13, 100, -32, 32, -100, -13, 57, 3, 57, -68, -8, 4, 0, 33, -21, 1, -5, 1, 7, -5, 1).

Column 4 of A055080.
First differences of A006148.
Cf. A002620, A005783, A005785, A005786, A055066.

Vec(G(4, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^20 - x^19 + 4*x^18 + 9*x^17 + 23*x^16 + 39*x^15 + 90*x^14 + 131*x^13 + 204*x^12 + 238*x^11 + 252*x^10 + 238*x^9 + 204*x^8 + 131*x^7 + 90*x^6 + 39*x^5 + 23*x^4 + 9*x^3 + 4*x^2 - x + 1)/((1 - x^4)^3*(1 - x^3)^4*(1 - x^2)^3*(1 - x)^5).
a(n) ~ n^14/2092278988800. - Stefano Spezia, Aug 08 2022
a(n) = n^14/2092278988800 + n^13/19926466560 + n^12/418037760 + n^11/14598144 + 689*n^10/522547200 + 253*n^9/13934592 + 2184839*n^8/11705057280 + 10313*n^7/6967296 + 2319707*n^6/250822656 + 1817221*n^5/39813120 + 2405336243*n^4/13795246080 + 151784975*n^3/306561024 + 93746545019*n^2/95103590400 + 924100468541*n/717352796160 + 1 + (n^2/486 + 5*n/162 + 233/2187)*floor(n/3) + (n^2/256 + 15*n/256 + 101/512)*floor(n/4) - (n^3/1458 + 7*n^2/486 + 22*n/243 + 356/2187)*floor((n+1)/3) + (n^5/122880 + 5*n^4/16384 + 125*n^3/24576 + 359*n^2/8192 + 10967*n/61440 + 8461/32768)*floor(n/2) + (n/256 + 15/512)*floor((n+1)/4). - Vaclav Kotesovec, Aug 09 2022

More terms from Vladeta Jovovic, Jun 03 2000

a(0)=1 prepended by Alois P. Heinz, Aug 08 2022

A005786 Number of 6-covers of an unlabeled n-set.

Original entry on oeis.org

1, 6, 43, 336, 2864, 25326, 223034, 1890123, 15115098, 112980937, 787320629, 5121184083, 31188412225, 178517111561, 964196387369, 4933278065881, 23997707450765, 111358094980387, 494444748602595, 2106504840061571

Offset: 0

N. J. A. Sloane

nonn

Number of 6 X n binary matrices with at least one 1 in every column up to row and column permutations.

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Vladeta Jovovic, Binary matrices up to row and column permutations

Column 6 of A055080.

Cf. A002620, A005783, A005784, A005785, A055066.

Vec(G(6, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

More terms from Vladeta Jovovic, Jun 12 2000

a(0)=1 prepended by Alois P. Heinz, Aug 08 2022

cp's OEIS Frontend

A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

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A005783 Number of 3-covers of an unlabeled n-set.

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A057968 Triangle T(n,k) of numbers of minimal 5-covers of an unlabeled n+5-set that cover k points of that set uniquely (k=5,..,n+5).

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A055066 Number of 7-covers of an unlabeled n-set.

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A005784 Number of 4-covers of an unlabeled n-set.

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A005786 Number of 6-covers of an unlabeled n-set.

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