A051118 -id:A051118 - 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.

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).

A056104 Number of 9-antichain covers of a labeled n-set.

Original entry on oeis.org

20, 1484110, 15936368770, 59961701958816, 121740972715475096, 167109117756164222210, 176340421320592288902670, 154794453668193645059165412, 118987888829384136293188343172

Offset: 6

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 29 2000

nonn

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.

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

Cf. A056046-A056049, A056052, A051112-A051118.

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

A084870 Number of 3-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 6, 28, 190, 1692, 16766, 166028, 1586430, 14580412, 129654526, 1123451628, 9544185470, 79881877532, 661135445886, 5425962250828, 44250287565310, 359161631645052, 2904756409742846, 23429320590259628, 188594431902253950

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (28,-315,1820,-5684,9072,-5760).

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/3!)*(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n).

G.f.: ( 1-26*x+265*x^2-1330*x^3+3340*x^4-3432*x^5 ) / ( (6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(8*x-1)*(5*x-1) ). - R. J. Mathar, Jul 08 2011

A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

Original entry on oeis.org

1, 3, 51, 4129, 1439381, 814788851, 395927618035, 155157302244381, 51960586962031617, 15663181302847575559, 4402571746033946222639, 1180812802393866826858193, 306839347397532891662028733

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

G. C. Greubel, Table of n, a(n) for n = 0..415
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Table[(1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n).

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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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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A056104 Number of 9-antichain covers of a labeled n-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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Mathematica

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A084870 Number of 3-multiantichains of an n-set.

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Magma

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A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

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