A047707 -id:A047707 - cp's OEIS frontend

A051115 Number of monotone Boolean functions of n variables with 7 mincuts.

0, 0, 0, 0, 0, 490, 1308270, 1085660748, 483349680164, 147791677696350, 35419166732721930, 7189973830216081696, 1298090729995668204288, 215276329320562758744210, 33531967207612008887673350

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
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, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051114, A051116, A051117, A051118.

A051116 Number of monotone Boolean functions of n variables with 8 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 115, 1613250, 4693213105, 5971431466764, 4657267944250425, 2654563364004395160, 1223795727111874798255, 485987045749653063943998, 173253367143529540187635315, 57037488183550191520963561230

Offset: 0

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

nonn

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
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, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051114, A051115, A051117, A051118.

A051117 Number of monotone Boolean functions of n variables with 9 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 20, 1484230, 15946757960, 60089234465176, 122281201867047920, 168329227672583040430, 178185327268349957044060, 156921594738520322214197672, 121014019160263331691800711500

Offset: 0

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
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, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051114, A051115, A051116, A051118.

A056046 Number of 3-antichain covers of a labeled n-set.

Original entry on oeis.org

0, 0, 0, 2, 56, 790, 8380, 76482, 638736, 5043950, 38390660, 285007162, 2079779416, 14995363110, 107204473740, 761823557042, 5390550296096, 38026057186270, 267656481977620, 1881017836414122, 13204444871932776, 92618543463601430, 649270263511862300

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jul 25 2000

			There are 2 3-antichain covers of a labeled 3-set: {{1},{2},{3}}, {{1,2},{1,3},{2,3}}.

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 = 0..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
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Cf. A047707.

Table[(1/6)*(7^n-6*5^n+6*4^n+3*3^n-6*2^n+2), {n, 0, 50}] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{22,-190,820,-1849,2038,-840},{0,0,0,2,56,790},30] (* Harvey P. Dale, Dec 09 2017 *)

for(n=0,50, print1((1/6)*(7^n-6*5^n+6*4^n+3*3^n-6*2^n+2), ", ")) \\ G. C. Greubel, Oct 06 2017

a(n) = (1/6)*(7^n-6*5^n+6*4^n+3*3^n-6*2^n+2).

G.f.: -2*x^3*(31*x^2-6*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Nov 27 2012

A059119 Triangle a(n,m)=number of m-element antichains on a labeled n-set; number of monotone n-variable Boolean functions with m mincuts (lower units), m=0..binomial(n,floor(n,2)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 1, 8, 9, 2, 1, 16, 55, 64, 25, 6, 1, 1, 32, 285, 1090, 2020, 2146, 1380, 490, 115, 20, 2, 1, 64, 1351, 14000, 82115, 304752, 759457, 1308270, 1613250, 1484230, 1067771, 635044, 326990, 147440, 57675, 19238, 5325, 1170, 190, 20, 1, 1

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

Row sums give A000372.

			[1, 1],
[1, 2],
[1, 4, 1],
[1, 8, 9, 2],
[1, 16, 55, 64, 25, 6, 1],
[1, 32, 285, 1090, 2020, 2146, 1380, 490, 115, 20, 2], ...

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

Suresh Govindarajan and Deepak Aditya, Table of n, a(n) for n = 0..94
K. S. Brown, Dedekind's Problem
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).

Cf. A000372, A016269, A047707, A051112-A051118.

a(n, 0) = 1; a(n, 1) = 2^n; a(n, 2) = A016269(n); a(n, 3) = A047707(n); a(n, 4) = A051112(n); a(5, n) = A051113(n); a(6, n) = A051114(n); a(7, n) = A051115(n); a(8, n) = A051116(n); a(9, n) = A051117(n); a(10, n) = A051118(n).

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

Original entry on oeis.org

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

A094036 Number of connected 5-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 6, 2005, 280971, 22795136, 1345702092, 65250058251, 2781911443317, 108660434574142, 3991349973006198, 140293749275697017, 4775521611056597583, 158758002632650598268, 5185922974307536588224

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

nonn

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

E.g.f.: (exp(31*x)-20*exp(23*x)+60*exp(19*x)+20*exp(17*x)
+5*exp(16*x)-105*exp(15*x)-120*exp(14*x)+150*exp(13*x)+180*exp(12*x)
-300*exp(11*x)-110*exp(10*x)+380*exp(9*x)+160*exp(8*x)-575*exp(7*x)
+570*exp(6*x)-186*exp(5*x)-975*exp(4*x)+1645*exp(3*x)-1030*exp(2*x)
+274*exp(x)-24)/5!.

A094034 Number of connected 3-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 1, 38, 645, 7510, 71981, 617358, 4947685, 37972070, 283229661, 2072354878, 14964711125, 107078983830, 761312910541, 5388481567598, 38017703680965, 267622831854790, 1880882526962621, 13203901505935518, 92616363612417205

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

With[{nmax = 50}, CoefficientList[Series[(Exp[7*x] - 6*Exp[5*x] + 3*Exp[4*x] + 14*Exp[3*x] - 21*Exp[2*x] + 11*Exp[x] - 2)/3!, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)
LinearRecurrence[{22,-190,820,-1849,2038,-840},{0,0,0,1,38,645,7510},30] (* Harvey P. Dale, Sep 20 2022 *)

x='x+O('x^50); concat([0,0,0], Vec(-x^3*(5*x+1)*(56*x^2-11*x-1)/( (x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)))) \\ G. C. Greubel, Oct 07 2017

E.g.f.: (exp(7*x) - 6*exp(5*x) + 3*exp(4*x) + 14*exp(3*x) - 21*exp(2*x) + 11*exp(x) -2)/3!.

G.f.: -x^3*(5*x+1)*(56*x^2-11*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Nov 27 2012

A094035 Number of connected 4-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 20, 1655, 65305, 1794730, 40179930, 793030245, 14423331635, 248261291960, 4113063835540, 66327037011235, 1049050826515965, 16360528085273190, 252545239130514350, 3869090307434050625, 58948119057416280295, 894447719738683138420

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..845

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

With[{nmax = 50}, CoefficientList[Series[(Exp[15*x] - 12*Exp[11*x] + 24*Exp[9*x] - 14*Exp[7*x] + 27*Exp[6*x] - 60*Exp[5*x] - 24*Exp[4*x] + 155*Exp[3*x] - 141*Exp[2*x] + 50*Exp[x] - 6)/4!, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0,0,0,0], Vec(serlaplace((exp(15*x) -12*exp(11*x) +24*exp(9*x) -14*exp(7*x) +27*exp(6*x) -60*exp(5*x) -24*exp(4*x) +155*exp(3*x) -141*exp(2*x) +50*exp(x) -6)/4!))) \\ G. C. Greubel, Oct 07 2017

concat(vector(4), Vec(5*x^4*(4+79*x-988*x^2-4414*x^3+52260*x^4-8721*x^5-374220*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)*(1-7*x)*(1-9*x)*(1-11*x)*(1-15*x)) + O(x^30))) \\ Colin Barker, Oct 13 2017

E.g.f.: (exp(15*x) - 12*exp(11*x) + 24*exp(9*x) - 14*exp(7*x) + 27*exp(6*x) - 60*exp(5*x) - 24*exp(4*x) + 155*exp(3*x) - 141*exp(2*x) + 50*exp(x) - 6)/4!.

G.f.: 5*x^4*(4+79*x-988*x^2-4414*x^3+52260*x^4-8721*x^5-374220*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)*(1-7*x)*(1-9*x)*(1-11*x)*(1-15*x)). - Colin Barker, Oct 13 2017

A056078 Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.

Original entry on oeis.org

0, 0, 2, 15, 54, 141, 306, 588, 1036, 1710, 2682, 4037, 5874, 8307, 11466, 15498, 20568, 26860, 34578, 43947, 55214, 68649, 84546, 103224, 125028, 150330, 179530, 213057, 251370, 294959, 344346, 400086, 462768, 533016, 611490, 698887, 795942, 903429, 1022162

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

Also number of 3 X 3 matrices with nonnegative integer entries with zero main diagonal and without zero rows or columns, such that sum of all entries is n. - Vladeta Jovovic, Sep 06 2006

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v. A proper hypergraph is a hypergraph without empty hyperedges or hyperedges containing all nodes. - Vladeta Jovovic, Sep 06 2006

			There are 15 proper T_1-hypergraphs with 3 nodes and 4 hyperedges: {{3},{3},{2},{1}}, {{3},{2},{2},{1}}, {{3},{2},{2,3},{1}}, {{3},{2},{1},{1}}, {{3},{2},{1},{1,3}}, {{3},{2},{1},{1,2}}, {{3},{2},{1,3},{1,2}}, {{3},{2,3},{1},{1,2}}, {{3},{2,3},{1,3},{1,2}}, {{2},{2,3},{1},{1,3}}, {{2},{2,3},{1,3},{1,2}}, {{2,3},{2,3},{1,3},{1,2}}, {{2,3},{1},{1,3},{1,2}}, {{2,3},{1,3},{1,3},{1,2}}, {{2,3},{1,3},{1,2},{1,2}}.

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 = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A056005, A047707.

[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120: n in [0..25]]; // G. C. Greubel, Oct 07 2017

Table[(n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,25, print1((n^4 + 20*n^3 + 35*n^2 - 140*n + 84)*n/120, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = C(n+5,5) -6*C(n+3,3) +6*C(n+2,2) +3*C(n+1,1) -6*C(n,0).
a(n+1) = ( n^4 +20*n^3 +35*n^2 -140*n +84 )*n/120.
From Colin Barker, Jul 11 2013: (Start)
a(n) = (-240+394*n-135*n^2-35*n^3+15*n^4+n^5)/120.
G.f.: x^3 *(x-2) *(2*x^2-2*x-1) / (x-1)^6. (End)

More terms from Colin Barker, Jul 11 2013

cp's OEIS Frontend

A051115 Number of monotone Boolean functions of n variables with 7 mincuts.

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A051116 Number of monotone Boolean functions of n variables with 8 mincuts.

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A051117 Number of monotone Boolean functions of n variables with 9 mincuts.

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A056046 Number of 3-antichain covers of a labeled n-set.

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A059119 Triangle a(n,m)=number of m-element antichains on a labeled n-set; number of monotone n-variable Boolean functions with m mincuts (lower units), m=0..binomial(n,floor(n,2)).

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

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A094036 Number of connected 5-element antichains on a labeled n-set.

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A094034 Number of connected 3-element antichains on a labeled n-set.

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A094035 Number of connected 4-element antichains on a labeled n-set.

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