A051114 -id:A051114 - cp's OEIS frontend

A051112 Number of monotone Boolean functions of n variables with 4 mincuts. Also Sperner systems with 4 blocks.

0, 0, 0, 0, 25, 2020, 82115, 2401910, 58089465, 1245331920, 24625121455, 460316430970, 8266174350005, 144171200793620, 2461016066613195, 41343340015862430, 686274244801356145, 11289648429330100120

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, #8, s(n,4).
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean function, Belgrade, 1999, in preparation.

Charles R Greathouse IV, Table of n, a(n) for n = 0..831 (next term has 1001 digits)
K. S. Brown, Dedekind's Problem
D. M. Cvetkovic, The number of antichains of finite power sets, Publ. Inst. Math., 13 (27), 1972, 5-9.
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 linear recurrences with constant coefficients, signature (82, -2970, 62700, -856713, 7947786, -51019100, 226259000, -678011136, 1304341632, -1445575680, 696729600).
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051113, A051114, A051115, A051116, A051117, A051118.

Table[(1/4!)*(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n-22*3^n+6*2^n),{n,0,20}] (* or *) LinearRecurrence[{82,-2970,62700,-856713,7947786,-51019100,226259000,-678011136,1304341632,-1445575680,696729600},{0,0,0,0,25,2020,82115,2401910,58089465,1245331920,24625121455},20] (* Harvey P. Dale, Nov 26 2019 *)

a(n)=(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n -22*3^n+6*2^n)/24 \\ Charles R Greathouse IV, Mar 14 2012

a(n) = (1/4!)*(16^n - 12*12^n + 24*10^n + 4*9^n - 18*8^n + 6*7^n - 36*6^n + 36*5^n + 11*4^n - 22*3^n + 6*2^n).
From Michael Somos: (Start)
a(n) = 82*a(n - 1) - 2970*a(n - 2) + 62700*a(n - 3) - 856713*a(n - 4) + 7947786*a(n - 5) - 51019100*a(n - 6) + 226259000*a(n - 7) - 678011136*a(n - 8) + 1304341632*a(n - 9) - 1445575680*a(n - 10) + 696729600*a(n - 11).
G.f.: 5x^4(5-6x-1855x^2+20076x^3-44356x^4-215280x^5+759168x^6) / ((1-3x)(1-4x)(1-5x)(1-6x)(1-2x)(1-7x)(1-8x)(1-9x)(1-10x)(1-12x)(1-16x)). (End)

A051118 Number of monotone Boolean functions of n variables with 10 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1067771, 43506231489, 501425871595264, 2719674203584968630, 9172837864705015158979, 22524989249381408262409893, 44328073635887914351462953684, 74381256243136645820404637874910

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, A051116, A051117.

A094033 Number of connected 2-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 3, 18, 75, 270, 903, 2898, 9075, 27990, 85503, 259578, 784875, 2366910, 7125303, 21425058, 64373475, 193317030, 580344303, 1741819338, 5227030875, 15684238350, 47059006503, 141189602418, 423593973075, 1270832250870

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

nonn

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 10 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 10.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Adam Roman, Igor T. Podolak and Agnieszka Deszynska, On the number of clusterings in a hierarchical classification model with overlapping clusters, Schedae Informaticae, Volume 20, 2011.
Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).

Cf. A016269, A047707.
Cf. A051112, A051113, A051114, A051115, A051116, A051117, A051118.
Cf. A094034, A094035, A094036, A094037.

[seq(stirling2(n,3)*3,n=0..26)]; # Zerinvary Lajos, Dec 06 2006

Table[3 StirlingS2[n, 3], {n, 0, 26}] (* Michael De Vlieger, Nov 30 2015 *)

x='x+O('x^50); concat([0,0,0],Vec(serlaplace((exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!))) \\ G. C. Greubel, Oct 06 2017

a(n) = 3 * A000392(n).
E.g.f.: (exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!.
From Colin Barker, Mar 31 2012: (Start)
a(n) = (3^n-3*2^n+3)/2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
G.f.: 3*x^3/((1-x)*(1-2*x)*(1-3*x)). (End)

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

Original entry on oeis.org

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.

A056049 Number of 6-antichain covers of a labeled n-set.

Original entry on oeis.org

1, 1375, 751192, 187216960, 29650991279, 3554308158345, 355235190457414, 31360944940860370, 2536696962910365277, 192628889065040142715, 13964833124133659520116, 978098391719401853480580

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 25 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.

Vincenzo Librandi, Table of n, a(n) for n = 4..200
K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers

Cf. A051114.

Table[(1 / 6!) (63^n - 30*47^n + 120*39^n + 60*35^n + 60 *33^n - 12*32^n - 345*31^n-720*29^n + 810*27^n + 120*26^n + 480*25^n + 360*24^n - 480*23^n - 720*22^n -240*21^n - 540*20^n + 1380*19^n + 750*18^n + 60*17^n - 210*16^n - 1535*15^n - 1820*14^n + 2250*13^n + 1800*12^n - 2820*11^n + 300*10^n + 2040*9^n + 340*8^n - 1815*7^n + 510*6^n - 1350*5^n + 1350*4^n + 274*3^n -548*2^n + 120), {n, 4, 20}] (* Vincenzo Librandi, Jun 17 2013 *)

a(n)=(1/6!)*(63^n - 30*47^n + 120*39^n + 60*35^n + 60*33^n - 12*32^n - 345*31^n - 720*29^n + 810*27^n + 120*26^n + 480*25^n + 360*24^n - 480*23^n - 720*22^n - 240*21^n - 540*20^n + 1380*19^n + 750*18^n + 60*17^n - 210*16^n - 1535*15^n - 1820*14^n + 2250*13^n + 1800*12^n - 2820*11^n + 300*10^n + 2040*9^n + 340*8^n - 1815*7^n + 510*6^n - 1350*5^n + 1350*4^n + 274*3^n - 548*2^n + 120).

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).

A056071 Number of 6-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 6 labeled nodes and n hyperedges.

Original entry on oeis.org

30, 8340, 780242, 29813578, 657271645, 10037038800, 117733967666, 1130702091428, 9273992351046, 66900184307860, 433616524985590, 2566055594813118, 14037125952339998, 71676448315103924, 344320192201127730, 1566076395413987110, 6779944255517707576

Offset: 4

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 26 2000

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.

K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers
Index entries for linear recurrences with constant coefficients, signature (64, -2016, 41664, -635376, 7624512, -74974368, 621216192, -4426165368, 27540584512, -151473214816, 743595781824, -3284214703056, 13136858812224, -47855699958816, 159518999862720, -488526937079580, 1379370175283520, -3601688791018080, 8719878125622720, -19619725782651120, 41107996877935680, -80347448443237920, 146721427591999680, -250649105469666120, 401038568751465792, -601557853127198688, 846636978475316672, -1118770292985239888, 1388818294740297792, -1620288010530347424, 1777090076065542336, -1832624140942590534, 1777090076065542336, -1620288010530347424, 1388818294740297792, -1118770292985239888, 846636978475316672, -601557853127198688, 401038568751465792, -250649105469666120, 146721427591999680, -80347448443237920, 41107996877935680, -19619725782651120, 8719878125622720, -3601688791018080, 1379370175283520, -488526937079580, 159518999862720, -47855699958816, 13136858812224, -3284214703056, 743595781824, -151473214816, 27540584512, -4426165368, 621216192, -74974368, 7624512, -635376, 41664, -2016, 64, -1).

Cf. A051114 for 6-element (unordered) antichains on a labeled n-element set, A056005.

a(n)=C(n + 63, 63) - 30*C(n + 47, 47) + 120*C(n + 39, 39) + 60*C(n + 35, 35) + 60*C(n + 33, 33) - 12*C(n + 32, 32) - 345*C(n + 31, 31) - 720*C(n + 29, 29) + 810*C(n + 27, 27) + 120*C(n + 26, 26) + 480*C(n + 25, 25) + 360*C(n + 24, 24) - 480*C(n + 23, 23) - 720*C(n + 22, 22) - 240*C(n + 21, 21) - 540*C(n + 20, 20) + 1380*C(n + 19, 19) + 750*C(n + 18, 18) + 60*C(n + 17, 17) - 210*C(n + 16, 16) - 1535*C(n + 15, 15) - 1820*C(n + 14, 14) + 2250*C(n + 13, 13) + 1800*C(n + 12, 12) - 2820*C(n + 11, 11) + 300*C(n + 10, 10) + 2040*C(n + 9, 9) + 340*C(n + 8, 8) - 1815*C(n + 7, 7) + 510*C(n + 6, 6) - 1350*C(n + 5, 5) + 1350*C(n + 4, 4) + 274*C(n + 3, 3) - 548*C(n + 2, 2) + 120*C(n + 1, 1).

More terms from Sean A. Irvine, Apr 14 2022

cp's OEIS Frontend

A051112 Number of monotone Boolean functions of n variables with 4 mincuts. Also Sperner systems with 4 blocks.

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A051118 Number of monotone Boolean functions of n variables with 10 mincuts.

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A094033 Number of connected 2-element antichains on a labeled n-set.

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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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A056049 Number of 6-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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A056071 Number of 6-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 6 labeled nodes and n hyperedges.

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