cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

Keywords

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

Keywords

References

  • 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.

Links

Crossrefs

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

Views

Author

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    [seq(stirling2(n,3)*3,n=0..26)]; # Zerinvary Lajos, Dec 06 2006
  • Mathematica
    Table[3 StirlingS2[n, 3], {n, 0, 26}] (* Michael De Vlieger, Nov 30 2015 *)
  • PARI
    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

Formula

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)

A051114 Number of monotone Boolean functions of n variables with 6 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1380, 759457, 192504214, 31169837405, 3827970163920, 392135190780649, 35468973527445018, 2937270598777421269, 228156280366446932500, 16904255174464832812001, 1208995011493806361868862, 84197134590686932418878093, 5746616155270206518199693720
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

Keywords

References

  • J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
  • V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

Links

Crossrefs

Cf. A016269, A047707, A051112, A051113, A051115, A051116, A051117, A051118.

Formula

a(n) = (1/6!)*(64^n-30 * 48^n+ 120 * 40^n+ 60 * 36^n+ 60 * 34^n-12 * 33^n-345 * 32^n-720 * 30^n+ 810 * 28^n+ 120 * 27^n+ 480 * 26^n+ 360 * 25^n-480 * 24^n-720 * 23^n-240 * 22^n-540 * 21^n+ 1380 * 20^n+ 750 * 19^n+ 60 * 18^n-210 * 17^n-1535 * 16^n-1820 * 15^n+ 2250 * 14^n+ 1800 * 13^n-2820 * 12^n+ 300 * 11^n+ 2040 * 10^n+ 340 * 9^n-1815 * 8^n+ 510 * 7^n-1350 * 6^n+ 1350 * 5^n+ 274 * 4^n-548 * 3^n+ 120 * 2^n).

Extensions

More terms from Colin Barker, Nov 26 2014

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

Keywords

References

  • 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.

Links

Crossrefs

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

Keywords

References

  • 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.

Links

Crossrefs

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

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

Keywords

Comments

Row sums give A000372.

Examples

			[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], ...

References

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

Links

Crossrefs

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

Formula

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).

A056101 Number of 8-antichain covers of a labeled n-set.

Original entry on oeis.org

115, 1612560, 4681922770, 5933930926484, 4603693881222819, 2608258836131076210
Offset: 5

Views

Author

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic, Jul 28 2000

Keywords

References

  • 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.

Links

Crossrefs

Cf. A056046-A056049, A056052, A051116.
Showing 1-8 of 8 results.

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