A032263 -id:A032263 - cp's OEIS frontend

A051304 Number of 4-element proper antichains of an n-element set.

0, 0, 0, 0, 5, 780, 41545, 1442910, 39400305, 923889960, 19550316665, 384954289170, 7196416532305, 129495073447740, 2264887575116985, 38775513868485030, 653195404307491505, 10869004241198535120, 179171681947204584505, 2932562923651659410490, 47737465871974206925905

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

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

Cf. A032263, A036239, A051112.

[(16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/(24): n in [0..50]]; // G. C. Greubel, Oct 07 2017

Table[(16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/4!, {n, 0, 50}] (* G. C. Greubel, Oct 07 2017 *)

for(n=0,50, print1((16^n - 18*12^n + 60*10^n - 9*9^n - 102*8^n + 105*7^n - 90*6^n + 95*5^n - 31*4^n - 33*3^n + 28*2^n - 6)/4!, ", ")) \\ G. C. Greubel, Oct 07 2017

a(n) = (1/4!) * (16^n -18*12^n +60*10^n -9*9^n -102*8^n +105*7^n -90*6^n +95*5^n -31*4^n -33*3^n +28*2^n -6).

G.f. x^4*( 5 +365*x -7935*x^2 +46885*x^3 -191420*x^4 +2285460*x^5 -14380560*x^6 +27216000*x^7 ) / ( (x-1) *(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) ). - R. J. Mathar, Jun 13 2013

Terms a(16) onward added by G. C. Greubel, Oct 07 2017

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

Original entry on oeis.org

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).

A133800 Triangle read by rows in which row n gives number of ways to partition n labeled elements into k pie slices allowing the pie to be turned over (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 3, 1, 15, 25, 30, 12, 1, 31, 90, 195, 180, 60, 1, 63, 301, 1050, 1680, 1260, 360, 1, 127, 966, 5103, 12600, 15960, 10080, 2520, 1, 255, 3025, 23310, 83412, 158760, 166320, 90720, 20160, 1, 511, 9330, 102315, 510300, 1369620

Offset: 1

Barry Cipra and N. J. A. Sloane, Jan 17 2008

			Triangle begins:
1,
1,  1,
1,  3,   1,
1,  7,   6,    3,
1, 15,  25,   30,   12,
1, 31,  90,  195,  180,   60,
1, 63, 301, 1050, 1680, 1260, 360.
...
For row n = 4 we have the following "pies":
. 1
./ \
2 . 3 . 12 .. 12 . 123
.\ / .. / \ .(..)..(..)
. 4 .. 3--4 . 34 .. 4 .. (1234)
k=4 .. k=3 ..k=2 . k=2 . k=1
(3)....(6)...(3)..(4)... (1)

C. G. Bower, Transforms (2)

Row sums give A032262. Diagonals give A000225, A000392, A032263, A133799, A001710.

A001710 := proc(n) if n < 2 then 1; else n!/2 ; fi ; end: A008277 := proc(n,k) combinat[stirling2](n,k) ; end: A133800 := proc(n,k) A008277(n,k)*A001710(k-1) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d, ",A133800(n,k)) ; od: od: # R. J. Mathar, Jan 18 2008

A001710[n_] := If[n<2, 1, n!/2]; A008277[n_, k_] := StirlingS2[n, k]; A133800[n_, k_] := A008277[n, k]*A001710[k-1]; Table[A133800[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014, after R. J. Mathar *)
(* A (n >= 0, k >= 0)-based version: *)
A133800[n_, k_] := k! StirlingS2[n+1, k+1] / If[k>1, 2, 1];
Table[A133800[n,k], {n,0,9}, {k,0,n}] // Flatten (* Peter Luschny, Oct 19 2017 *)

Take triangle of Stirling numbers of second kind (A008277) and multiply k-th column by A001710(k) (order of alternating group A_k).

More terms from R. J. Mathar, Jan 18 2008

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).

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).

A134165 Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 3) x = y.

Original entry on oeis.org

1, 3, 8, 24, 86, 348, 1478, 6324, 26846, 112668, 467798, 1925124, 7867406, 31980588, 129475718, 522603924, 2104600766, 8461122108, 33972973238, 136278002724, 546271650926

Offset: 0

Ross La Haye, Jan 12 2008

			a(2) = 8 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{},{1}}, {{},{2}}, {{},{1,2}} and we have for case 1 {{1},{2}} and we have for case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 2.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000225, A000392, A032263, A000079.

LinearRecurrence[{10,-35,50,-24},{1,3,8,24},30] (* Harvey P. Dale, Feb 29 2020 *)

a(n) = (1/2)(4^n - 2*3^n + 5*2^n - 2) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + 2*StirlingS2(n+1,2) + 1.

G.f.: (1-7*x+13*x^2-x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). [Colin Barker, Jul 30 2012]

A134168 Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y.

Original entry on oeis.org

1, 3, 9, 30, 111, 438, 1779, 7290, 29871, 121998, 496299, 2011650, 8129031, 32769558, 131850819, 529745610, 2126058591, 8525561118, 34166421339, 136858609170, 548013994551, 2193796224678, 8780408783859, 35137313082330, 140596298752911, 562526359448238, 2250528981434379, 9003386657325090

Offset: 0

Ross La Haye, Jan 12 2008

			a(2) = 9 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{},{1}}, {{},{2}}, {{},{1,2}} and we have for case 2 {{1},{1,2}}, {{2},{1,2}} and we have for case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000225, A032263, A028243, A000079.

LinearRecurrence[{10, -35, 50, -24}, {1, 3, 9, 30}, 50] (* or *) Table[(1/2)*(4^n - 3^n + 3*2^n - 1), {n,0,50}] (* G. C. Greubel, May 30 2016 *)

a(n) = (1/2)*(4^n - 3^n + 3*2^n - 1).
a(n) = 3*StirlingS2(n+1,4) +2*StirlingS2(n+1,3) +2*StirlingS2(n+1,2) +1.
G.f.: -(5*x^3 - 14*x^2 + 7*x - 1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Jul 30 2012

A133789 Let P(A) denote the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.

Original entry on oeis.org

0, 1, 4, 16, 70, 316, 1414, 6196, 26590, 112156, 466774, 1923076, 7863310, 31972396, 129459334, 522571156, 2104535230, 8460991036, 33972711094, 136277478436, 546270602350, 2188566048076, 8764718254054, 35090241492916, 140455083984670, 562102715143516

Offset: 0

Ross La Haye, Jan 03 2008, Jan 08 2008

nonn

Also, number of even binomial coefficient in rows 0 to 2^n of Pascal's triangle. [Aaron Meyerowitz, Oct 29 2013]

			a(3) = 16 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we see that
{1} and {2},
{1} and {3},
{2} and {3},
{1} and {2,3},
{2} and {1,3},
{3} and {1,2}
are disjoint, while
{} and {1},
{} and {2},
{} and {3},
{} and {1,2},
{} and {1,3},
{} and {2,3},
{} and {1,2,3}
are disjoint and one is a subset of the other and
{1,2} and {1,3},
{1,2} and {2,3},
{1,3} and {2,3}
are intersecting, but neither is a subset of the other.
Also, through row 8 of Pascal's triangle the a(3)=16 even entries are 2 (so a(0)=0 and a(1)=1) then 4,6,4 (so a(2)=4) then 10,10 then  6,20,6 then 8,28,56,70,56,28,8. [_Aaron Meyerowitz_, Oct 29 2013]

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [_Ross La Haye_, Feb 22 2009]

Cf. A082134, A002697, A020522, A006516, A007582, A000302.

Cf. A000225, A000392, A032263.

a(n) = (1/2)(4^n - 2*3^n + 3*2^n - 2).
O.g.f.: x*(1-6*x+11*x^2)/[(-1+x)*(-1+2*x)*(-1+3*x)*(-1+4*x)]. - R. J. Mathar, Jan 11 2008
a(n) = A084869(n)-1 = A016269(n-2)+2^n-1. - Vladeta Jovovic, Jan 04 2008, corrected by Eric Rowland, May 15 2017
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). [Aaron Meyerowitz, Oct 29 2013]

Edited by N. J. A. Sloane, Jan 20 2008 to incorporate suggestions from several contributors.

A134018 Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x.

Original entry on oeis.org

0, 1, 3, 10, 45, 226, 1113, 5230, 23565, 102826, 438273, 1836550, 7601685, 31183426, 127084233, 515429470, 2083077405, 8396552026, 33779262993, 135696871990, 544528258725, 2183337968626, 8749031918553, 35043178292110, 140313885993645, 561679104393226

Offset: 0

Ross La Haye, Jan 10 2008

			a(3) = 10 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we have for case 0 {{},{1}}, {{},{2}}, {{},{3}}, {{},{1,2}}, {{},{1,3}}, {{},{2,3}}, {{},{1,2,3}} and we have for case 1 {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [_Ross La Haye_, Feb 22 2009]
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000225, A032263.

LinearRecurrence[{10,-35,50,-24},{0,1,3,10},30] (* Harvey P. Dale, Dec 01 2017 *)

a(n) = (1/2)(4^n - 3^(n+1) + 5*2^n - 3) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,2).

G.f.: x*(1-7*x+15*x^2)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). [Colin Barker, Jul 29 2012]

More terms from Harvey P. Dale, Dec 01 2017

cp's OEIS Frontend

A051304 Number of 4-element proper antichains of an n-element set.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

A133800 Triangle read by rows in which row n gives number of ways to partition n labeled elements into k pie slices allowing the pie to be turned over (n >= 1, 1 <= k <= n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Views