A051180 -id:A051180 - cp's OEIS frontend

A053157 Number of 3-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

1, 5, 32, 235, 1816, 14055, 107052, 800315, 5886416, 42739855, 307295572, 2193374595, 15571898616, 110121224855, 776543100092, 5464689616075, 38398915520416, 269529406433055, 1890416947176612, 13251578251332755

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

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).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Cf. A051180.

[(7^n -3*5^n +3*4^n +2*3^n -3*2^n +2)/6: n in [1..50]]; // G. C. Greubel, Oct 07 2017

Table[(7^n -3*5^n +3*4^n +2*3^n -3*2^n +2)/6, {n,1,50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=1,50, print1((7^n -3*5^n +3*4^n +2*3^n -3*2^n +2)/6, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = (7^n -3*5^n +3*4^n +2*3^n -3*2^n +2)/6.

G.f.: -x*(280*x^5-475*x^4+339*x^3-112*x^2+17*x-1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Jul 29 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

Vladeta Jovovic, Goran Kilibarda

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 0..550

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

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

Vladeta Jovovic, Goran Kilibarda

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 0..660

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

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 *)

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

Vladeta Jovovic, Goran Kilibarda

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).

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

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

[(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

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 *)

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

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

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

Vladeta Jovovic, Goran Kilibarda

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 0..550

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

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 *)

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

Vladeta Jovovic, Goran Kilibarda

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 0..660

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

[(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

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 *)

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

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).

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 15, 30, 30, 5, 1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1, 1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 28 2000

An antichain is called intersecting (or proper) antichain if every two members have a nonempty intersection. Row sums give the number of intersecting antichains on a labeled n-set or n-variable Boolean functions in the Post class F(7,2) or self-dual monotone Boolean functions of n+1 variables. Cf. A001206.

			1;
1, 1;
1, 3;
1, 7, 3, 1;
1, 15, 30, 30, 5;
1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1;
1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7;

Jovovic V., Kilibarda G., The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
Pogosyan G., Miyakawa M., Nozaki A., Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Cf. A001206, A032263, A051303-A051307, A036239, A051180-A051185, A016269, A047707, A051112-A051118, A000372.

T(n, 0)=1, T(n, 1)=2^n-1, T(n, 2)=A032263(n), T(n, 3)=A051303(n), T(n, 4)=A051304(n), T(n, 5)=A051305(n), T(n, 6)=A051306(n), T(n, 7)=A051307(n).

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

Vladeta Jovovic, Goran Kilibarda

nonn

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).

Index entries for sequences related to Boolean functions

A051184(n)-A051183(n)+A051182(n)-A051181(n)+A051180(n)-A036239(n)+A000918(n)

cp's OEIS Frontend

A053157 Number of 3-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

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

Views

Author

Keywords