A051112 -id:A051112 - 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).

A051303 Number of 3-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 1, 30, 605, 9030, 110901, 1200150, 11932285, 111885510, 1006471301, 8786447670, 75039565965, 630534185190, 5234341175701, 43059373189590, 351805681631645, 2859550165976070, 23152657123816101, 186907026783617910, 1505512392025329325

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).

Cf. A032263, A036239, A051112.

[(8^n - 9*6^n + 15*5^n - 4*4^n - 9*3^n + 8*2^n - 2) / Factorial(3) : n in [0..25]]; // G. C. Greubel, Oct 06 2017

A051303:=n->(8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!: seq(A051303(n), n=0..30); # Wesley Ivan Hurt, Oct 06 2017

Table[(8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!, {n,0,25}] (* G. C. Greubel, Oct 06 2017 *)

for(n=0,25, print1((8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2 )/3!, ", ")) \\ G. C. Greubel, Oct 06 2017

a(n) = (8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!.
G.f.: x^3*(360*x^3-78*x^2-x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Nov 27 2012
a(n) = 29*a(n-1) - 343*a(n-2) + 2135*a(n-3) - 7504*a(n-4) + 14756*a(n-5) - 14832*a(n-6) + 5760*a(n-7) for n > 6. - Wesley Ivan Hurt, Oct 06 2017
E.g.f.: exp(x)*(exp(x) - 1)^3*(2 - 2*exp(x) - 3*exp(2*x) + 3*exp(3*x) + exp(4*x))/6. - Stefano Spezia, Sep 28 2024

A056069 Number of 4-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 4 labeled nodes and n hyperedges.

Original entry on oeis.org

25, 454, 3818, 21420, 92805, 335152, 1055944, 2990020, 7767357, 18789070, 42797602, 92588216, 191542842, 381000192, 731941256, 1363109096, 2468549141, 4358716470, 7520830306, 12706161124, 21054530855, 34269633840, 54863015040, 86489873580, 134406530985

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Jul 26 2000

nonn

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.

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.

T. D. Noe, Table of n, a(n) for n = 4..1000
K. S. Brown, Dedekind's problem

Cf. A051112 for 4-element (unordered) antichains on a labeled n-element set, A056005.

a(n) = C(n + 15, 15) - 12*C(n + 11, 11) + 24*C(n + 9, 9) + 4*C(n + 8, 8) - 18*C(n + 7, 7) + 6*C(n + 6, 6) - 36*C(n + 5, 5) + 36*C(n + 4, 4) + 11*C(n + 3, 3) - 22*C(n + 2, 2) + 6*C(n + 1, 1).

Empirical G.f.: x^4*(6*x^10 -62*x^9 +271*x^8 -636*x^7 +800*x^6 -328*x^5 -495*x^4 +812*x^3 -446*x^2 +54*x +25)/(x-1)^16. [Colin Barker, May 29 2012]

A056090 Number of 4-element ordered antichain covers of an unlabeled n-element set.

Original entry on oeis.org

25, 429, 3364, 17602, 71385, 242347, 720792, 1934076, 4777337, 11021713, 24008532, 49790614, 98954626, 189457350, 350941064, 631167840, 1105440045, 1890167329, 3162113836, 5185330818, 8348369731, 13215102985, 20593381200, 31626858540, 47916657405, 71681161365

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Jul 27 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.

G. C. Greubel, Table of n, a(n) for n = 4..1000
K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Cf. A056047 for 4-antichain (unordered) covers of a labeled n-set, A051112. See also A056074, A056093.

[(-104270181120*n + 236073062016*n^2 - 169534943760*n^3 + 28403538800*n^4 + 12862329480*n^5 - 2983956976*n^6 - 613678065*n^7 + 39763295*n^8 + 21456435*n^9 + 2461459*n^10 + 143325*n^11 + 5005*n^12 + 105*n^13 + n^14)/Factorial(14): n in [4..50]]; // G. C. Greubel, Oct 06 2017

Table[(-104270181120 n + 236073062016 n^2 - 169534943760 n^3 + 28403538800 n^4 + 12862329480 n^5 - 2983956976 n^6 - 613678065 n^7 + 39763295 n^8 + 21456435 n^9 + 2461459 n^10 + 143325 n^11 + 5005 n^12 + 105 n^13 + n^14)/(14)!, {n, 4, 50}] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1},{25,429,3364,17602,71385,242347,720792,1934076,4777337,11021713,24008532,49790614,98954626,189457350,350941064},30] (* Harvey P. Dale, Dec 09 2021 *)

for(n=4,50, print1((-104270181120*n + 236073062016*n^2 - 169534943760*n^3 + 28403538800*n^4 + 12862329480*n^5 - 2983956976*n^6 - 613678065*n^7 + 39763295*n^8 + 21456435*n^9 + 2461459*n^10 + 143325*n^11 + 5005*n^12 + 105*n^13 + n^14)/(14)!, ", ")) \\ G. C. Greubel, Oct 06 2017

a(n) = C(n + 14, 14) - 12*C(n + 10, 10) + 24*C(n + 8, 8) + 4*C(n + 7, 7) - 18*C(n + 6, 6) + 6*C(n + 5, 5) - 36*C(n + 4, 4) + 36*C(n + 3, 3) + 11*C(n + 2, 2) - 22*C(n + 1, 1) + 6*C(n, 0).
G.f.: x^4*(6*x^10 -62*x^9 +271*x^8 -636*x^7 +800*x^6 -328*x^5 -495*x^4 +812*x^3 -446*x^2 +54*x +25)/(1-x)^15. - Colin Barker, May 29 2012
a(n) = (-104270181120 n + 236073062016 n^2 - 169534943760 n^3 + 28403538800 n^4 + 12862329480 n^5 - 2983956976 n^6 - 613678065 n^7 + 39763295 n^8 + 21456435 n^9 + 2461459 n^10 + 143325 n^11 + 5005 n^12 + 105 n^13 + n^14)/(14)!. - G. C. Greubel, Oct 06 2017

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!.

A051304 Number of 4-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 5, 780, 41545, 1442910, 39400305, 923889960, 19550316665, 384954289170, 7196416532305, 129495073447740, 2264887575116985, 38775513868485030, 653195404307491505, 10869004241198535120, 179171681947204584505, 2932562923651659410490, 47737465871974206925905

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A032263, A036239, A051112.

[(16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/(24): n in [0..50]]; // G. C. Greubel, Oct 07 2017

Table[(16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/4!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,50, print1((16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/4!, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = (1/4!) * (16^n -18*12^n +60*10^n -9*9^n -102*8^n +105*7^n -90*6^n +95*5^n -31*4^n -33*3^n +28*2^n -6).

G.f. x^4*( 5 +365*x -7935*x^2 +46885*x^3 -191420*x^4 +2285460*x^5 -14380560*x^6 +27216000*x^7 ) / ( (x-1) *(9*x-1) *(6*x-1) *(7*x-1) *(3*x-1) *(5*x-1) *(2*x-1) *(12*x-1) *(10*x-1) *(4*x-1) *(8*x-1) *(16*x-1) ). - R. J. Mathar, Jun 13 2013

Terms a(16) onward added by G. C. Greubel, Oct 07 2017

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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A051303 Number of 3-element proper antichains of an n-element set.

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A056069 Number of 4-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 4 labeled nodes and n hyperedges.

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A056090 Number of 4-element ordered antichain covers of an unlabeled n-element set.

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Magma

Mathematica

PARI

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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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A051304 Number of 4-element proper antichains of an n-element set.

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