A051112 -id:A051112 - cp's OEIS frontend

A051305 Number of 5-element proper antichains of an n-element set.

0, 0, 0, 0, 0, 543, 118629, 12564636, 907001550, 51751693161, 2527016053023, 110737868741742, 4489929936371880, 171944175793168779, 6309813148166785257, 224210698542088771968

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

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

Cf. A032263, A036239, A051112.

[(32^n - 30*24^n + 150*20^n - 45*18^n + 85*17^n - 515*16^n -450*15^n + 1365*14^n + 390*13^n - 1680*12^n - 22*11^n + 1875*10^n - 1080*9^n - 685*8^n + 980*7^n - 669*6^n + 575*5^n - 195*4^n - 150*3^n + 124*2^n - 24)/(120): n in [0..50]]; // G. C. Greubel, Oct 07 2017

Table[(32^n - 30*24^n + 150*20^n - 45*18^n + 85*17^n - 515*16^n -450*15^n + 1365*14^n + 390*13^n - 1680*12^n - 22*11^n + 1875*10^n - 1080*9^n - 685*8^n + 980*7^n - 669*6^n + 575*5^n - 195*4^n - 150*3^n + 124*2^n - 24)/5!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,50, print1((32^n - 30*24^n + 150*20^n - 45*18^n + 85*17^n - 515*16^n -450*15^n + 1365*14^n + 390*13^n - 1680*12^n - 22*11^n + 1875*10^n - 1080*9^n - 685*8^n + 980*7^n - 669*6^n + 575*5^n - 195*4^n - 150*3^n + 124*2^n - 24)/5!, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = (1/5!)*(32^n -30*24^n +150*20^n -45*18^n +85*17^n -515*16^n -450*15^n +1365*14^n +390*13^n -1680*12^n -22*11^n +1875*10^n -1080*9^n -685*8^n +980*7^n -669*6^n +575*5^n -195*4^n -150*3^n +124*2^n -24).

A051306 Number of 6-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 0, 300, 233821, 78501094, 15532759830, 2213672795040, 254206334062527, 25146386270836578, 2235664320306737320, 183782806231396191820, 14248056393984957136593

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

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

Cf. A032263, A036239, A051112.

Table[(64^n - 45*48^n + 300*40^n - 135*36^n + 510*34^n - 198*33^n - 1499*32^n - 2700*30^n + 6615*28^n + 1215*27^n - 780*26^n + 3750*25^n - 6750*24^n - 8280*23^n + 3828*22^n - 12285*21^n + 19425*20^n + 31635*19^n - 30105*18^n - 34425*17^n + 24770*16^n + 13125*15^n - 3885*14^n + 390*13^n - 5670*12^n - 12485*11^n + 28575*10^n - 16560*9^n - 3435*8^n + 7868*7^n - 4995*6^n + 3800*5^n - 1301*4^n - 822*3^n + 668*2^n - 120)/6!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

a(n) = (1/6!)*(64^n -45*48^n +300*40^n -135*36^n +510*34^n -198*33^n -1499*32^n -2700*30^n +6615*28^n +1215*27^n -780*26^n +3750*25^n -6750*24^n -8280*23^n +3828*22^n -12285*21^n +19425*20^n +31635*19^n -30105*18^n -34425*17^n +24770*16^n +13125*15^n -3885*14^n +390*13^n -5670*12^n -12485*11^n +28575*10^n -16560*9^n -3435*8^n +7868*7^n -4995*6^n +3800*5^n -1301*4^n -822*3^n +668*2^n -120).

A094034 Number of connected 3-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 1, 38, 645, 7510, 71981, 617358, 4947685, 37972070, 283229661, 2072354878, 14964711125, 107078983830, 761312910541, 5388481567598, 38017703680965, 267622831854790, 1880882526962621, 13203901505935518, 92616363612417205

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

With[{nmax = 50}, CoefficientList[Series[(Exp[7*x] - 6*Exp[5*x] + 3*Exp[4*x] + 14*Exp[3*x] - 21*Exp[2*x] + 11*Exp[x] - 2)/3!, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)
LinearRecurrence[{22,-190,820,-1849,2038,-840},{0,0,0,1,38,645,7510},30] (* Harvey P. Dale, Sep 20 2022 *)

x='x+O('x^50); concat([0,0,0], Vec(-x^3*(5*x+1)*(56*x^2-11*x-1)/( (x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)))) \\ G. C. Greubel, Oct 07 2017

E.g.f.: (exp(7*x) - 6*exp(5*x) + 3*exp(4*x) + 14*exp(3*x) - 21*exp(2*x) + 11*exp(x) -2)/3!.

G.f.: -x^3*(5*x+1)*(56*x^2-11*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Nov 27 2012

A094035 Number of connected 4-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 20, 1655, 65305, 1794730, 40179930, 793030245, 14423331635, 248261291960, 4113063835540, 66327037011235, 1049050826515965, 16360528085273190, 252545239130514350, 3869090307434050625, 58948119057416280295, 894447719738683138420

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..845

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

With[{nmax = 50}, CoefficientList[Series[(Exp[15*x] - 12*Exp[11*x] + 24*Exp[9*x] - 14*Exp[7*x] + 27*Exp[6*x] - 60*Exp[5*x] - 24*Exp[4*x] + 155*Exp[3*x] - 141*Exp[2*x] + 50*Exp[x] - 6)/4!, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0,0,0,0], Vec(serlaplace((exp(15*x) -12*exp(11*x) +24*exp(9*x) -14*exp(7*x) +27*exp(6*x) -60*exp(5*x) -24*exp(4*x) +155*exp(3*x) -141*exp(2*x) +50*exp(x) -6)/4!))) \\ G. C. Greubel, Oct 07 2017

concat(vector(4), Vec(5*x^4*(4+79*x-988*x^2-4414*x^3+52260*x^4-8721*x^5-374220*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)*(1-7*x)*(1-9*x)*(1-11*x)*(1-15*x)) + O(x^30))) \\ Colin Barker, Oct 13 2017

E.g.f.: (exp(15*x) - 12*exp(11*x) + 24*exp(9*x) - 14*exp(7*x) + 27*exp(6*x) - 60*exp(5*x) - 24*exp(4*x) + 155*exp(3*x) - 141*exp(2*x) + 50*exp(x) - 6)/4!.

G.f.: 5*x^4*(4+79*x-988*x^2-4414*x^3+52260*x^4-8721*x^5-374220*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)*(1-7*x)*(1-9*x)*(1-11*x)*(1-15*x)). - Colin Barker, Oct 13 2017

A051307 Number of 7-element proper antichains of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 0, 135, 329205, 365924948, 205640068950, 75013516525425, 20611869786684495, 4661763066154503606, 917701003163074793520, 163180081989646991509955, 26889766005753182579964345, 4182467653250525215771670424, 622388054953695081193665509610

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

Cf. A032263, A036239, A051112.

a(n) = 1/7! * (128^n -63*96^n +525*80^n -315*72^n +1785*68^n -1386*66^n +455*65^n -3486*64^n -9450*60^n +21315*56^n +8505*54^n -13650*52^n -5355*51^n +36750*50^n -5145*49^n -14805*48^n -57960*46^n -4725*45^n +45738*44^n +36120*43^n -191835*42^n +43050*41^n +74725*40^n -73710*39^n +333165*38^n +104895*37^n -73395*36^n -54390*35^n -354144*34^n -423192*33^n +383621*32^n +143220*31^n -292425*30^n +753855*29^n +181545*28^n -314685*27^n -114660*26^n -916125*25^n -268716*24^n +1998493*23^n +140833*22^n -2359350*21^n +458675*20^n +2147950*19^n -961758*18^n -1428000*17^n +933380*16^n +578175*15^n -614362*14^n +143052*13^n +45990*12^n -244860*11^n +356475*10^n -199521*9^n -12244*8^n +64778*7^n -40026*6^n +28035*5^n -9604*4^n -5292*3^n +4248*2^n -720).

A056047 Number of 4-antichain covers of a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 25, 1895, 70370, 1868650, 41062035, 802349205, 14514339340, 249104207000, 4120588431245, 66392465654515, 1049608974433110, 16365222591176550, 252584307401055655, 3869412829938587825, 58950765174112191680, 894469325684769169300, 13531152125348360663265

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jul 25 2000

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 = 0..845
K. S. Brown, Dedekind's problem
Eric Weisstein's World of Mathematics, Antichain covers

Cf. A051112.

[(15^n - 12*11^n + 24*9^n + 4*8^n - 18*7^n + 6*6^n - 36*5^n + 36*4^n + 11*3^n - 22*2^n + 6)/24: n in [0..25]]; // G. C. Greubel, Oct 07 2017

Table[(1/4!)*(15^n - 12*11^n + 24*9^n + 4*8^n - 18*7^n + 6*6^n - 36*5^n + 36*4^n + 11*3^n - 22*2^n + 6), {n,0,25}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,25, print1((15^n - 12*11^n + 24*9^n + 4*8^n - 18*7^n + 6*6^n - 36*5^n + 36*4^n + 11*3^n - 22*2^n + 6)/24, ", ")) \\ G. C. Greubel, Oct 07 2017

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

G.f.: -5*x^4*(517752*x^6 -251184*x^5 +4757*x^4 +12696*x^3 -1810*x^2 +24*x +5) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(11*x -1)*(15*x -1)). - Colin Barker, Jul 11 2013

More terms from Colin Barker, Jul 11 2013

A084870 Number of 3-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 6, 28, 190, 1692, 16766, 166028, 1586430, 14580412, 129654526, 1123451628, 9544185470, 79881877532, 661135445886, 5425962250828, 44250287565310, 359161631645052, 2904756409742846, 23429320590259628, 188594431902253950

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (28,-315,1820,-5684,9072,-5760).

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/3!)*(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n).

G.f.: ( 1-26*x+265*x^2-1330*x^3+3340*x^4-3432*x^5 ) / ( (6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(8*x-1)*(5*x-1) ). - R. J. Mathar, Jul 08 2011

A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

Original entry on oeis.org

1, 3, 51, 4129, 1439381, 814788851, 395927618035, 155157302244381, 51960586962031617, 15663181302847575559, 4402571746033946222639, 1180812802393866826858193, 306839347397532891662028733

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

G. C. Greubel, Table of n, a(n) for n = 0..415
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Table[(1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n).

A037843 Number of matrices with n columns whose rows do not cover each other; ordered antichains of subsets of an n-set.

Original entry on oeis.org

2, 3, 7, 39, 2551, 22928343, 6641112790058484007

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jul 23 2000

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
Index entries for sequences related to Boolean functions

Cf. A003182, A051112-A051117, A007411, A006602.

a(n)=Sum_{k=0..C(n, floor(n/2))}k!*M(n, k) where M(n, k) is the number of distinct monotone Boolean functions of n variables with k mincuts.

A056163 Number of ordered antichains on an unlabeled n-set; labeled T_1-hypergraphs with n hyperedges.

Original entry on oeis.org

2, 3, 5, 11, 120, 191297

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jul 31 2000

A 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.

			a(1)=1+2=3; a(2)=1+3+1=5; a(3)=1+4+4+2=11; a(4)=1+5+10+19+25+30+30=120; a(5)=1+6+20+90+454+2206+8340+20580+38640+60480+60480=191297.
There are 11 ordered antichains on an unlabeled 3-set: 0, (0), ({1}), ({1,2}), ({1,2,3}), ({1},{2}), ({1},{2,3}), ({2,3},{1}), ({1,2},{1,3}), ({1},{2},{3}), ({1,2},{1,3},{2,3}).

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. A000372 for (unordered) antichains on a labeled n-set, A056005, A056069-A056071, A056073, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

a(n)=Sum_{k=0..C(n, floor(n/2))}b(k, n) where b(k, n) is the number of k-element ordered antichains on an unlabeled n-set.

cp's OEIS Frontend

A051305 Number of 5-element proper antichains of an n-element set.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A051306 Number of 6-element proper antichains of an n-element set.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A094034 Number of connected 3-element antichains on a labeled n-set.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094035 Number of connected 4-element antichains on a labeled n-set.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A051307 Number of 7-element proper antichains of an n-element set.

Views

Author

Keywords

Crossrefs

Formula

A056047 Number of 4-antichain covers of a labeled n-set.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A084870 Number of 3-multiantichains of an n-set.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs