A056090 -id:A056090 - cp's OEIS frontend

A056074 Number of 3-element ordered antichain covers of an unlabeled n-element set.

2, 17, 71, 212, 518, 1106, 2142, 3852, 6534, 10571, 16445, 24752, 36218, 51716, 72284, 99144, 133722, 177669, 232883, 301532, 386078, 489302, 614330, 764660, 944190, 1157247, 1408617, 1703576

Offset: 3

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

G. C. Greubel, Table of n, a(n) for n = 3..1000
K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A056046 for 3-antichain (unordered) covers of a labeled n-set, A047707. See also A056090, A056093.

[n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720: n in [3..25]]; // G. C. Greubel, Oct 06 2017

A056074:=n->n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720: seq(A056074(n), n=3..60); # Wesley Ivan Hurt, Oct 06 2017

LinearRecurrence[{7,-21,35,-35,21,-7,1},{2,17,71,212,518,1106,2142},30] (* or *) Table[Binomial[n+6,6]-6Binomial[n+4,4]+6Binomial[n+3,3]+ 3Binomial[n+2,2]- 6Binomial[n+1,1]+ 2Binomial[n,0],{n,3,30}] (* Harvey P. Dale, Jul 12 2011 *)

a(n)=n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720 \\ Charles R Greathouse IV, Feb 19 2017

a(n) = C(n + 6, 6) - 6*C(n + 4, 4) + 6*C(n + 3, 3) + 3*C(n + 2, 2) - 6*C(n + 1, 1) + 2*C(n, 0).
a(0)=2, a(1)=17, a(2)=71, a(3)=212, a(4)=518, a(5)=1106, a(6)=2142, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Jul 12 2011
G.f.: (-2-3*x+6*x^2-2*x^3)/(x-1)^7. - Harvey P. Dale, Jul 12 2011
a(n) = n*(n^5 + 21*n^4 - 5*n^3 - 345*n^2 + 724*n - 396)/720. - Charles R Greathouse IV, Feb 19 2017

A056093 Number of 5-element ordered antichain covers of an unlabeled n-element set.

Original entry on oeis.org

30, 2176, 54036, 709956, 6290051, 42606671, 237197942, 1135834242, 4823607212, 18563958502, 65783057592, 217240417628, 674884181813, 1987124979703, 5579019610088, 15010371955248, 38862554420034, 97163223921924, 235290234202584, 553296290481584

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 27 2000

nonn

V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

G. C. Greubel, Table of n, a(n) for n = 4..1000
K. S. Brown, Dedekind's problem
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
Eric Weisstein's World of Mathematics, Antichain covers

Cf. A056048 for 5-antichain (unordered) covers of a labeled n-set, A051113. See also A056074, A056090.

Table[Binomial[n+30,30]-20 Binomial[n+22,22]+60 Binomial[n+18,18]+ 20 Binomial[n+16,16]+ 10 Binomial[n+15,15]-110 Binomial[n+14,14]- 120 Binomial[n+13,13]+ 150 Binomial[n+12,12]+ 120 Binomial[n+11,11]- 240 Binomial[n+10,10]+ 20 Binomial[n+9,9]+ 240 Binomial[n+8,8]+ 40 Binomial[n+7,7]- 205 Binomial[n+6,6]+ 60 Binomial[n+5,5]- 210 Binomial[n+4,4]+ 210 Binomial[n+3,3]+ 50 Binomial[n+2,2]- 100 Binomial[n+1,1]+ 24 Binomial[n,0],{n,4,30}] (* Harvey P. Dale, Sep 06 2011 *)

a(n)=C(n + 30, 30) - 20*C(n + 22, 22) + 60*C(n + 18, 18) + 20*C(n + 16, 16) + 10*C(n + 15, 15) - 110*C(n + 14, 14) - 120*C(n + 13, 13) + 150*C(n + 12, 12) + 120*C(n + 11, 11) - 240*C(n + 10, 10) + 20*C(n + 9, 9) + 240*C(n + 8, 8) + 40*C(n + 7, 7) - 205*C(n + 6, 6) + 60*C(n + 5, 5) - 210*C(n + 4, 4) + 210*C(n + 3, 3) + 50*C(n + 2, 2) - 100*C(n + 1, 1) + 24*C(n, 0).

More terms from Harvey P. Dale, Sep 06 2011

A056164 Number of ordered antichain covers of an unlabeled n-set; labeled T_1-hypergraphs (without empty hyperedges) with n hyperedges.

Original entry on oeis.org

1, 2, 6, 109, 191177

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Jul 31 2000

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

			There are 6 ordered antichain covers on an unlabeled 3-set: ({1,2,3}), ({1},{2,3}), ({2,3},{1}), ({1,2},{1,3}), ({1},{2},{3}), ({1,2},{1,3},{2,3}).
a(3)=1+3+2=6; a(4)=1+6+17+25+30+30=109; a(5)=1+10+71+429+2176+8310+20580+38640+60480+60480=191177.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers

Cf. A056074, A056090, A056093, A000372, A056005, A056069-A056071, A056073, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

a(n)=Sum_{k=1..C(n, floor(n/2))}b(k, n) where b(k, n) is the number of k-element ordered antichains covers of an unlabeled n-set.

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A056074 Number of 3-element ordered antichain covers of an unlabeled n-element set.

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A056093 Number of 5-element ordered antichain covers of an unlabeled n-element set.

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A056164 Number of ordered antichain covers of an unlabeled n-set; labeled T_1-hypergraphs (without empty hyperedges) with n hyperedges.

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