A051118 -id:A051118 - cp's OEIS frontend

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

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.

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).

A084871 Number of 4-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 7, 41, 398, 6177, 128232, 2881531, 62769238, 1288737197, 25012685732, 463681018671, 8294783320578, 144410750517217, 2462999084589232, 41359616334934211, 686406989350511918, 11290725888842193237

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..825
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

[(16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!.
From R. J. Mathar, Jul 08 2011: (Start)
G.f.: (-1 + 80*x - 2813*x^2 + 57293*x^3 - 749139*x^4 + 6577949*x^5 - 39353597*x^6 + 158972472*x^7 - 417774220*x^8 + 651991536*x^9 - 465379200*x^10) / ( (9*x-1) *(6*x-1) *(7*x-1) *(3*x-1) *(5*x-1) *(2*x-1) *(12*x-1) *(10*x-1) *(4*x-1) *(8*x-1) *(16*x-1) ).
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). (End)

A084872 Number of 5-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 8, 56, 726, 17938, 722680, 35955180, 1798971434, 83885891894, 3612380896332, 145277787750064, 5534505187364062, 202229611397865690, 7158136006402746464, 247316732670273773108, 8389241054998193347410

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..660
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

[(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/5!)*(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n).

A084873 Number of 6-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 9, 73, 1212, 44667, 3251186, 345094227, 39552733796, 4234657495267, 409948262617398, 36190736880911571, 2964860272283578040, 229165985114590010307, 16940021231116707830570

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..550
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

Table[(1/6!)*(64^n - 30*48^n + 120*40^n + 60*36^n + 60*34^n - 12*33^n - 315*32^n - 720*30^n + 810*28^n + 120*27^n + 480*26^n + 360*25^n - 1080*24^n - 720*23^n - 240*22^n - 540*21^n + 3180*20^n + 750*19^n + 660*18^n + 90*17^n - 4535*16^n - 5420*15^n + 6750*14^n + 5400*13^n - 13620*12^n + 900*11^n + 16440*10^n + 2740*9^n - 12165*8^n + 4110*7^n - 25650*6^n + 25650*5^n + 10474*4^n - 20948*3^n + 7560*2^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/6!)*(64^n - 30*48^n + 120*40^n + 60*36^n + 60*34^n - 12*33^n - 315*32^n - 720*30^n + 810*28^n + 120*27^n + 480*26^n + 360*25^n - 1080*24^n - 720*23^n - 240*22^n - 540*21^n + 3180*20^n + 750*19^n + 660*18^n + 90*17^n - 4535*16^n - 5420*15^n + 6750*14^n + 5400*13^n - 13620*12^n + 900*11^n + 16440*10^n + 2740*9^n - 12165*8^n + 4110*7^n - 25650*6^n + 25650*5^n + 10474*4^n - 20948*3^n + 7560*2^n).

A084874 Number of (k,m,n)-antichains of multisets with k=3 and m=2.

Original entry on oeis.org

0, 0, 9, 162, 2025, 21870, 219429, 2112642, 19847025, 183642390, 1682955549, 15327821322, 139038251625, 1257873017310, 11360034454869, 102475388237202, 923689006041825, 8321664254958630, 74945757885541389, 674816499677616282

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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

a(n) is also the number of entries that are divisible by 3 in rows 0 through 3^n-1 of Pascal's triangle A007318. - Tim Cieplowski, Nov 25 2014

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 (18,-99,162).

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

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

Table[(9^n - 2*6^n + 3^n)/2, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{18,-99,162},{0,0,9},20] (* Harvey P. Dale, Oct 01 2023 *)

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

a(n) = (1/2!)*(9^n - 2*6^n + 3^n).
G.f.: -9*x^2 / ( (6*x-1)*(3*x-1)*(9*x-1) ). - R. J. Mathar, Jul 08 2011
E.g.f.: (exp(9*x) - 2*exp(6*x) + exp(3*x))/2. - G. C. Greubel, Oct 08 2017

A084875 Number of (k,m,n)-antichains of multisets with k=3 and m=3.

Original entry on oeis.org

0, 0, 1, 350, 24025, 1061570, 38306701, 1238697950, 37547263825, 1093418309690, 31035659056501, 866306577308150, 23915774118612025, 655397866616830610, 17872808187862527901, 485794481046271815950, 13175146525408965630625

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. 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..695
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (77,-2277,32895,-242514,854388,-1102248).

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

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

Table[(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{77,-2277,32895,-242514,854388,-1102248},{0,0,1,350,24025,1061570},20] (* Harvey P. Dale, May 29 2025 *)

for(n=0,50, print1((27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/3!)*(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n).

G.f.: -x^2*(-1-273*x+648*x^2+24300*x^3) / ( (18*x-1)*(9*x-1)*(6*x-1)*(3*x-1)*(14*x-1)*(27*x-1) ). - R. J. Mathar, Jul 08 2011

A084876 Number of (k,m,n)-antichains of multisets with k=3 and m=4.

Original entry on oeis.org

0, 0, 0, 310, 159300, 32389900, 4469327850, 503689260970, 50466655894200, 4701945998612200, 418104908350395750, 36055756736065208230, 3046399249526576159700, 253883533322134812268900, 20963248884482293139928450, 1720141562616331422239725090

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. 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..520
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

[(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n)/24, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n)/24, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/4!)*(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n).

G.f.: -10*x^3*(-31 - 5173*x + 663390*x^2 - 16812297*x^3 - 320866029*x^4 + 19383439320*x^5 - 243502067160*x^6 + 252158125680*x^7 + 6816687418800*x^8) / ( (6*x-1) *(54*x-1) *(42*x-1) *(3*x-1) *(9*x-1) *(27*x-1) *(31*x-1) *(26*x-1) *(18*x-1) *(81*x-1) *(36*x-1) *(14*x-1) ). - R. J. Mathar, Jul 08 2011

A084877 Number of (k,m,n)-antichains of multisets with k=3 and m=5.

Original entry on oeis.org

0, 0, 0, 114, 649850, 678772108, 377819587984, 153135104560046, 51758494975477206, 15644366957608679376, 4400899140179858419388, 1180668574169021790713938, 306827161657039584492179842

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. 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 + 10*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 120*54^n + 240*42^n + 40*36^n - 240*31^n + 35*27^n + 60*26^n - 210*18^n + 210*14^n + 50*9^n - 100*6^n + 24*3^n), {n, 0, 1000}] (* 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 + 10*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 120*54^n + 240*42^n + 40*36^n - 240*31^n + 35*27^n + 60*26^n - 210*18^n + 210*14^n + 50*9^n - 100*6^n + 24*3^n).

A084878 Number of (k,m,n)-antichains of multisets with k=3 and m=6.

Original entry on oeis.org

0, 0, 0, 15, 1729366, 10340309701, 24380294253318, 36539301527565851, 42407896071362952494, 42091311943805278602897, 37781049596189171124466966, 31727275407315883994852626087

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. 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..345
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

a(n) = (1/6!)*(729^n - 30*486^n + 120*378^n + 60*324^n + 60*294^n - 360*279^n - 12*276^n - 720*252^n + 15*243^n + 90*234^n + 720*231^n + 120*216^n + 720*210^n - 240*205^n + 360*196^n - 720*189^n - 180*187^n + 720*186^n - 720*176^n + 120*168^n - 720*167^n + 360*165^n - 300*162^n - 720*157^n + 180*156^n + 720*148^n - 240*145^n + 720*138^n + 30*134^n - 240*129^n + 900*126^n - 360*120^n + 180*111^n + 300*108^n - 20*102^n + 150*98^n - 1800*93^n - 1800*84^n + 85*81^n + 450*78^n + 1800*77^n + 1800*70^n - 1800*63^n + 300*56^n - 1020*54^n + 2040*42^n + 340*36^n - 2040*31^n + 225*27^n + 510*26^n - 1350*18^n + 1350*14^n + 274*9^n - 548*6^n + 120*3^n).

cp's OEIS Frontend

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

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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).

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A084871 Number of 4-multiantichains of an n-set.

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A084872 Number of 5-multiantichains of an n-set.

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A084873 Number of 6-multiantichains of an n-set.

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A084874 Number of (k,m,n)-antichains of multisets with k=3 and m=2.

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A084875 Number of (k,m,n)-antichains of multisets with k=3 and m=3.

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A084876 Number of (k,m,n)-antichains of multisets with k=3 and m=4.

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