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 1-7 of 7 results.

A051185 Number of intersecting families of an n-element set. Also number of n-variable clique Boolean functions.

Original entry on oeis.org

2, 6, 40, 1376, 1314816, 912818962432, 291201248266450683035648, 14704022144627161780744368338695925293142507520, 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

Comments

Also the number of n-ary Boolean polymorphisms of the binary Boolean relation OR, namely the Boolean functions f(x1,...,xn) with the property that (x1 or y1) and ... and (xn or yn) implies f(x1,...,xn) or f(y1,...,yn). - Don Knuth, Dec 04 2019
These values are necessarily divisible by powers of 2. The sequence of exponents begins 1, 1, 3, 5, 12, 22, 49, 93, ... , 2^(n-1)-C(n-1,floor(n/2)-1), ... (cf. A191391). - Andries E. Brouwer, Aug 07 2012
a(1) = 2^1.
a(2) = 6 = 2^1 * 3
a(3) = 2^3 * 5.
a(4) = 2^5 * 43.
a(5) = 2^12 * 3 * 107.
a(6) = 2^22 * 13 * 16741.
a(7) = 2^49 * 2111 * 245039,
a(8) = 2^93 * 3^2 * 5 * 7211 * 76697 * 59656829,
a(9) = 2^200 * 1823 * 2063 * 576967 * 3600144350906020591.
An intersecting family is a collection of subsets of {1,2,...,n} such that the intersection of every subset with itself or with any other subset in the family is nonempty. The maximum number of subsets in an intersecting family is 2^(n-1). - Geoffrey Critzer, Aug 16 2013

Examples

			a(2) = 6 because we have: {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {1, 2}}, {{2}, {1, 2}}. - _Geoffrey Critzer_, Aug 16 2013

References

  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
  • Pogosyan G., Miyakawa M., A. Nozaki, Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Links

Crossrefs

Cf. A036239, A051180-A051184.

Programs

  • Mathematica
    Table[Length[
  Select[Subsets[Subsets[Range[1, n]]],
   Apply[And,
     Flatten[Table[
       Table[Intersection[#[[i]], #[[j]]] != {}, {i, 1,
Length[#]}], {j, 1, Length[#]}]]] &]], {n, 1, 4}] (* Geoffrey Critzer, Aug 16 2013 *)

Extensions

a(8)-a(9) by Andries E. Brouwer, Aug 07 2012, Dec 11 2012

A051363 Number of 6-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 112, 40286, 5485032, 534844548, 45066853496, 3538771308282, 267882021563464, 19861835713621616, 1453175611052688600, 105278656040052332838, 7564280930105061931496, 539399446172552069053404
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Crossrefs

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Formula

a(n) = (1/6!)*(64^n -20*56^n +90*52^n +30*50^n +25*49^n -420*48^n -180*47^n +450*46^n +60*45^n +615*44^n +1683*43^n -3252*42^n -6030*41^n +8520*40^n +10560*39^n -15849*38^n -13005*37^n +26410*36^n +10655*35^n -50385*34^n +33390*33^n +29480*32^n -82010*31^n +91215*30^n -67380*29^n +36870*28^n -15249*27^n +4380*26^n -1215*25^n +1390*24^n -695*23^n -1574*22^n +3255*21^n -3075*20^n +1800*19^n -675*18^n +150*17^n +70*16^n -340*14^n +510*13^n -340*12^n +85*11^n -225*8^n +225*7^n +274*4^n -274*3^n -120*2^n +120).

A051364 Number of 5-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 225, 21571, 1174122, 51441824, 2038356243, 76714338477, 2804947403364, 100732231517698, 3572491367063421, 125474030774355263, 4371052010746528926, 151172238539268318372
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Crossrefs

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Programs

  • Mathematica
    Table[1/5! (32^n - 10*28^n + 30*26^n + 5*25^n - 80*24^n + 45*23^n + 105*22^n - 217*21^n + 205*20^n - 120*19^n + 45*18^n - 10*17^n - 9*16^n + 40*14^n - 60*13^n + 40*12^n - 10*11^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

Formula

a(n) = (1/5!)*(32^n - 10*28^n + 30*26^n + 5*25^n - 80*24^n + 45*23^n + 105*22^n - 217*21^n + 205*20^n - 120*19^n + 45*18^n - 10*17^n - 9*16^n + 40*14^n - 60*13^n + 40*12^n - 10*11^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24).

A051365 Number of 4-element families of an n-element set such that every 3 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 3, 275, 8475, 192385, 3831093, 71466675, 1285857975, 22632300245, 392522268633, 6734698919575, 114576024346875, 1935649374363705, 32505459713369373, 543014736097852475, 9029329231317194175, 149522990698790644765, 2466942184607949641313
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Crossrefs

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Programs

  • Magma
    [(16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[1/4! (16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6)/24, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/4!)*(16^n - 4*14^n + 6*13^n - 4*12^n + 11^n - 6*8^n + 6*7^n + 11*4^n - 11*3^n - 6*2^n + 6).
G.f.: -x^3*(47062848*x^7 -42816008*x^6 +13976678*x^5 -2170583*x^4 +168932*x^3 -5672*x^2 +2*x +3) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(7*x -1)*(8*x -1)*(11*x -1)*(12*x -1)*(13*x -1)*(14*x -1)*(16*x -1)). - Colin Barker, Jul 12 2013

Extensions

More terms from Colin Barker, Jul 12 2013

A051366 Number of 6-element families of an n-element set such that every 4 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 112, 39761, 5318420, 506289623, 41378309308, 3133123494417, 227657567966500, 16152548751321851, 1129224692910819164, 78169242144478858373, 5373159786842137703140, 367368738925063893430959
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Crossrefs

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Programs

  • Mathematica
    Table[1/6! (64^n - 15*60^n + 60*58^n + 25*57^n - 240*56^n + 45*55^n + 705*54^n - 987*53^n - 351*52^n + 3040*51^n - 5445*50^n + 6105*49^n - 4939*48^n + 2997*47^n - 1365*46^n + 455*45^n - 105*44^n + 15*43^n - 42^n - 15*32^n + 75*30^n - 150*29^n + 150*28^n - 75*27^n + 15*26^n + 85*16^n - 85*15^n - 225*8^n + 225*7^n + 274*4^n - 274*3^n - 120*2^n + 120), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

Formula

a(n) = (1/6!)*(64^n - 15*60^n + 60*58^n + 25*57^n - 240*56^n + 45*55^n + 705*54^n - 987*53^n - 351*52^n + 3040*51^n - 5445*50^n + 6105*49^n - 4939*48^n + 2997*47^n - 1365*46^n + 455*45^n - 105*44^n + 15*43^n - 42^n - 15*32^n + 75*30^n - 150*29^n + 150*28^n - 75*27^n + 15*26^n + 85*16^n - 85*15^n - 225*8^n + 225*7^n + 274*4^n - 274*3^n - 120*2^n + 120).

A051367 Number of 5-element families of an n-element set such that every 4 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 224, 21281, 1144027, 49310674, 1915317642, 70460566827, 2513684751809, 88008877380908, 3043421159408080, 104321464544910613, 3552122530256316471, 120307381384305672102
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Crossrefs

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Programs

  • Magma
    [(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24)/120: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(1/5!)*(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((1/5!)*(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24), ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/5!)*(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24).

A051368 Number of Boolean functions of n variables and rank 8 from the Post class F(5,2).

Original entry on oeis.org

0, 0, 0, 12, 105765, 59046810, 16636450912, 3491313542424, 627725748292995, 102894277877828670, 15867914519581210614, 2343602605748557069356, 335205287948366997151705, 46782266953279485879549090
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • E. Post, Two-valued iterative systems, Annals of Mathematics, no 5, Princeton University Press, NY, 1941.
  • 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 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Links

Formula

A051184(n)-A051183(n)+A051182(n)-A051181(n)+A051180(n)-A036239(n)+A000918(n)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.