A059048 -id:A059048 - cp's OEIS frontend

A059052 Number of n-element unlabeled ordered T_0-antichains; T_1-hypergraphs (with empty hyperedge but without multiple hyperedges) on n labeled nodes.

2, 4, 4, 72, 38040, 4020463392, 18438434825136728352, 340282363593610211921722192165556850240, 115792089237316195072053288318104625954343609704705784618785209431974668731584

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

nonn

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

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

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

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.

Cf. A059048, A059049, A059050, A059051, A326960.

a(n) = Sum_{m=0..2^n} A(n, m), where A(n, m) is number of n-element ordered T_0-antichains on an unlabeled m-set. Cf. A059048.

a(n) = row sums of A059048.

A059083 Number of T_0-antichains on a labeled n-set.

Original entry on oeis.org

2, 3, 3, 8, 96, 6373, 7725703, 2414518872815, 56130437161078967568912

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point.

			a(0) = 1 + 1, a(1) = 1 + 2, a(2) = 2 + 1, a(3) = 6 + 2, a(4) = 12 + 52 + 25 + 6 + 1, a(5) = 520 + 1770 + 2086 + 1370 + 490 + 115 + 20 + 2.

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

V. Jovovic, 3-element T_0-antichains on a labeled 4-set
V. Jovovic, Formula for the number of m-element T_0-antichains on a labeled n-set
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).

Cf. A059079-A059082, A059048-A059052.

Cf. A000372.

a(n) = Sum_{m=0..binomial(n, floor(n/2))} A(m, n), where A(m, n) is number of m-element T_0-antichains on a labeled n-set. Cf. A059080.

a(n) = column sums of A059080.

More terms from Vladeta Jovovic, Nov 28 2003

A059049 Number of 6-element ordered T_0-antichains on an unlabeled n-set; T_1-hypergraphs on 6 labeled nodes with n (not necessarily empty) distinct hyperedges (n=0,1,...,64).

Original entry on oeis.org

0, 0, 0, 0, 30, 8220, 738842, 25256626, 464670831, 5570534392, 48655319306, 332222541564, 1859009659336, 8811850222304, 36244568422086, 131710639199900, 428697293437675, 1263065928235140, 3396450715952370

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

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.

Sean A. Irvine, Table of n, a(n) for n = 0..64

Cf. A059048, A059050-A059052.

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

A059050 Number of 7-element ordered T_0-antichains on an unlabeled n-set; T_1-hypergraphs on 7 labeled nodes with n (not necessarily empty) distinct hyperedges (n=0,1,...,128).

Original entry on oeis.org

0, 0, 0, 0, 0, 20580, 9106440, 1058272828, 56671435220, 1819138009252, 40526220292026, 685404291322800, 9333711208757096, 106588763704012184, 1051025434717631806, 9144977489478933912, 71381946761468630363

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

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.

Sean A. Irvine, Table of n, a(n) for n = 0..128

Cf. A059048, A059049, A059051, A059052.

a(n)=C(128, n) - 42*C(96, n) + 210*C(80, n) + 140*C(72, n) + 210*C(68, n) - 84*C(66, n) + 14 *C(65, n) - 819*C(64, n) - 2520*C(60, n) + 2730*C(56, n) + 840*C(54, n) + 840*C(52, n) - 420*C(51, n) + 2940*C(50, n) + 630*C(48, n) - 5040*C(46, n) + 840*C(45, n) - 1260*C(44, n) + 1680*C(43, n) - 9660*C(42, n) + 1260*C(41, n) + 3360*C(40, n) - 7560*C(39, n) + 11130*C(38, n) + 5880*C(37, n) + 9240*C(36, n) + 2982*C(35, n) - 6300*C(34, n) - 8652 *C(33, n) - 9905*C(32, n) - 8400*C(31, n) - 8540*C(30, n) + 13860*C(29, n) + 14490 *C( 28, n) - 5040*C(27, n) + 10500*C(26, n) + 10080*C(25, n) - 8120*C(24, n) - 15050*C(23, n) - 5040*C(22, n) - 11340*C(21, n) + 20580*C(20, n) + 15750*C(19, n) - 1540*C(18, n) - 5810*C(17, n) - 16485*C(16, n) - 21420*C(15, n) + 26250*C(14, n) + 21000*C(13, n) - 29820*C(12, n) + 3500*C(11, n) + 17640*C(10, n) + 2940*C(9, n) - 16016*C(8, n) + 4410*C(7, n) - 9744*C(6, n) + 9744*C(5, n) + 1764*C(4, n) - 3528*C(3, n) + 720*C(2, n).

A059079 Number of n-element T_0-antichains on a labeled set.

Original entry on oeis.org

2, 5, 19, 16654, 2369110564675, 5960531437586238714806902334250676, 479047836152505670895481840783987408043359908583921478726185296900312296071642855730299

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 23 2000

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point.

			a(0) = (1/0!)*[1!*e] = 2; a(1) = (1/1!)*[2!*e] = 5; a(2) = (1/2!)*([4!*e] - 2*[3!*e] + [2!*e]) = 19; a(3) = (1/3!)*([8!*e] - 6*[6!*e] + 6*[5!*e] + 3*[4!*e] - 6*[3!*e] + 2*[2!*e]) = 16654, where [n!*e]=floor(n!*exp(1)).

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.

Vladeta Jovovic, Illustration

Cf. A059080-A059083, A059048-A059052, A000522.

A059080 Triangle A(n,m) of numbers of n-element T_0-antichains on a labeled m-set, m=0,...,2^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 0, 1, 6, 12, 0, 0, 0, 2, 52, 520, 2640, 6720, 6720, 0, 0, 0, 0, 25, 1770, 53940, 1012620, 13487040, 136745280, 1094688000, 7025356800, 36084787200, 145297152000, 435891456000, 871782912000, 871782912000

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 29 2000

nonn

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. Row sums give A059079. Column sums give A059083.

			[1, 1], [1, 2, 2], [0, 0, 1, 6, 12], [0, 0, 0, 2, 52, 520, 2640, 6720, 6720], ...; there are 2 3-element T_0-antichains on a 3-set: {{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.

V. Jovovic, 3-element T_0-antichains on a labeled 4-set
V. Jovovic, Formula for the number of m-element T_0-antichains on a labeled n-set

Cf. A059079, A059081-A059083, A059048-A059052.

A059051 Number of ordered T_0-antichains on an unlabeled n-set; labeled T_1-hypergraphs with n (not necessarily empty) distinct hyperedges.

Original entry on oeis.org

2, 3, 2, 4, 99, 190866

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

			a(0) = 1 + 1, a(1) = 1 + 2, a(2) = 1 + 1, a(3) = 2 + 2, a(4) = 1 + 13 + 25 + 30 + 30, a(5) = 26 + 354 + 2086 + 8220 + 20580 + 38640 + 60480 + 60480.

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

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.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.

Cf. A059048-A059050, A059052.

a(n) = Sum_{m=0..C(n, floor(n/2))} A(m, n), where A(m, n) is number of m-element ordered T_0-antichains on an unlabeled n-set. Cf. A059048.

a(n) = column sums of A059048.

A059081 Number of 5-element T_0-antichains on a labeled n-set, n=0,..,32.

Original entry on oeis.org

0, 0, 0, 0, 6, 2086, 273072, 19371912, 940055760, 35289051840, 1099827892800, 29723466326400, 716351882400000, 15683016533184000, 315722887044364800, 5890186860509952000, 102288867798813696000, 1656523525703574528000

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point.

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..32
Vladeta Jovovic, Formula for the number of m-element T_0-antichains on a labeled n-set

Cf. A059079, A059080, A059082, A059083, A059048-A059052.

P[x_, n_] := (-1)^n*Pochhammer[-x, n]; Table[(1/5!)*(P[32, n] - 20*P[24, n] + 60*P[20, n] + 20*P[18, n] + 10*P[17, n] - 110*P[16, n] - 120*P[15, n] + 150*P[14, n] + 120*P[13, n] - 240*P[12, n] + 20*P[11, n] + 240*P[10, n] + 40*P[9, n] - 205*P[8, n] + 60*P[7, n] - 210*P[6, n] + 210*P[5, n] + 50*P[4, n] - 100*P[3, n] + 24*P[2, n]), {n, 0, 32}] (* G. C. Greubel, Oct 07 2017 *)

a(n) = (1/5!)*([32]_n - 20*[24]_n + 60*[20]_n + 20*[18]_n + 10*[17]_n - 110*[16]_n - 120*[15]_n + 150*[14]_n + 120*[13]_n - 240*[12]_n + 20*[11]_n + 240*[10]_n + 40*[9]_n - 205*[8]_n + 60*[7]_n - 210*[6]_n + 210*[5]_n + 50*[4]_n - 100*[3]_n + 24*[2]_n), where [k]_n := k*(k - 1)*...*(k - n + 1), [k]_0 = 1.

A059082 Number of 6-element T_0-antichains on a labeled n-set, n = 0, ..., 64.

Original entry on oeis.org

0, 0, 0, 0, 1, 1370, 738842, 176796382, 26021566536, 2807549333568, 245222809302240, 18418417704308160, 1236761946163054080, 76210520306627266560, 4388527139331858082560, 239214759548062858560000, 12457699161320493400320000, 623967599346727576292352000

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point.

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.

Alois P. Heinz, Table of n, a(n) for n = 0..64
V. Jovovic, Formula for the number of m-element T_0-antichains on a labeled n-set

Cf. A059079, A059080, A059081, A059083, A059048, A059049, A059050, A059051, A059052.

f:=proc(k,n) if k+1<=n then RETURN(0) else RETURN(k!/(k - n)!) fi: end;a:=n->(1/6!)*(f(64,n) - 30*f(48,n) + 120*f(40,n) + 60*f(36,n) + 60*f(34,n)- 12*f(33,n) - 345*f(32,n) - 720*f(30,n) + 810*f(28,n) + 120*f(27,n) + 480*f(26,n) + 360*f(25,n) - 480*f(24,n) - 720*f(23,n) - 240*f(22,n) - 540*f(21,n) + 1380*f(20,n) + 750*f(19,n) + 60*f(18,n) - 210*f(17,n) - 1535*f(16,n) - 1820*f(15,n) + 2250*f(14,n) + 1800*f(13,n) - 2820*f(12,n) + 300*f(11,n) + 2040*f(10,n) + 340*f(9,n) - 1815*f(8,n) + 510*f(7,n) - 1350*f(6,n) + 1350*f(5,n) + 274*f(4,n) - 548*f(3,n) + 120*f(2,n));seq(a(n),n=0..20); # Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005

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), where [k]_n := k*(k - 1)*...*(k - n + 1), [k]_0 = 1.

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005

cp's OEIS Frontend

A059052 Number of n-element unlabeled ordered T_0-antichains; T_1-hypergraphs (with empty hyperedge but without multiple hyperedges) on n labeled nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A059083 Number of T_0-antichains on a labeled n-set.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A059049 Number of 6-element ordered T_0-antichains on an unlabeled n-set; T_1-hypergraphs on 6 labeled nodes with n (not necessarily empty) distinct hyperedges (n=0,1,...,64).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A059050 Number of 7-element ordered T_0-antichains on an unlabeled n-set; T_1-hypergraphs on 7 labeled nodes with n (not necessarily empty) distinct hyperedges (n=0,1,...,128).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A059079 Number of n-element T_0-antichains on a labeled set.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A059080 Triangle A(n,m) of numbers of n-element T_0-antichains on a labeled m-set, m=0,...,2^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A059051 Number of ordered T_0-antichains on an unlabeled n-set; labeled T_1-hypergraphs with n (not necessarily empty) distinct hyperedges.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A059081 Number of 5-element T_0-antichains on a labeled n-set, n=0,..,32.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A059082 Number of 6-element T_0-antichains on a labeled n-set, n = 0, ..., 64.