A048143
Number of labeled connected simplicial complexes with n nodes.
Original entry on oeis.org
1, 1, 1, 5, 84, 6348, 7743728, 2414572893530, 56130437190053299918162
Offset: 0
For n=3 we could have 2 edges (in 3 ways), 3 edges (1 way), or 3 edges and a triangle (1 way), so a(3)=5.
a(5) = 1+75+645+1655+2005+1345+485+115+20+2 = 6348.
- Patrick De Causmaecker, Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
- Greg Huber, Letters to N. J. A. Sloane, May 1983 [Annotated, corrected, scanned copy]
- Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
- Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
A094037
Number of connected 6-element antichains on a labeled n-set.
Original entry on oeis.org
0, 0, 0, 0, 1, 1345, 738741, 185165477, 29458046177, 3541242666045, 354515664467077, 31326419674855789, 2535191648955942273, 192567615994193565125, 13962461827318220986133, 978010022290154153870661
Offset: 0
A094729
Number of connected ordered 2-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 7, 37, 151, 541, 1807, 5797, 18151, 55981, 171007, 519157, 1569751, 4733821, 14250607, 42850117, 128746951, 386634061, 1160688607, 3483638677, 10454061751, 31368476701, 94118013007, 282379204837, 847187946151, 2541664501741, 7625194831807
Offset: 0
-
With[{nmax = 50}, CoefficientList[Series[Exp[3*x] - 3*Exp[2*x] + 4*Exp[x] - 2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{6,-11,6},{0,1,1,7},30] (* Harvey P. Dale, Aug 07 2023 *)
-
x='x+O('x^50); concat([0], Vec(serlaplace(exp(3*x)-3*exp(2*x) +4*exp(x)-2))) \\ G. C. Greubel, Oct 06 2017
-
concat(0, Vec(x*(1 - 5*x + 12*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Oct 13 2017
A094738
Number of connected 6-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 26, 702, 34746, 2873097, 317812783, 36594544008, 3875472781976, 368569834860663, 31872207293370225, 2555189550184175334, 193269748160593198186, 13986349926952570806549, 978803975916211424325827
Offset: 0
-
E:= (1/6!)*(exp(63*x) - 30*exp(47*x) + 120*exp(39*x) + 60*exp(35*x) + 60*exp(33*x) - 18*exp(32*x) - 309*exp(31*x) - 720*exp(29*x) + 810*exp(27*x) + 120*exp(26*x) + 480*exp(25*x) + 480*exp(24*x) - 1200*exp(23*x) - 720*exp(22*x) - 240*exp(21*x) - 900*exp(20*x) + 3540*exp(19*x) + 615*exp(18*x) + 780*exp(17*x) + 585*exp(16*x) - 4295*exp(15*x) - 6870*exp(14*x) + 6210*exp(13*x) + 10020*exp(12*x) - 15960*exp(11*x) - 6510*exp(10*x) + 21960*exp(9*x) + 11610*exp(8*x) - 32715*exp(7*x) + 31185*exp(6*x) - 23670*exp(5*x) - 51405*exp(4*x) + 132334*exp(3*x) - 112152*exp(2*x) + 44304*exp(x) - 7560):
S:= series(E,x,21):
seq(coeff(S,x,i),i=0..20); # Robert Israel, Jul 14 2019
A094036
Number of connected 5-element antichains on a labeled n-set.
Original entry on oeis.org
0, 0, 0, 0, 6, 2005, 280971, 22795136, 1345702092, 65250058251, 2781911443317, 108660434574142, 3991349973006198, 140293749275697017, 4775521611056597583, 158758002632650598268, 5185922974307536588224
Offset: 0
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
-
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
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
-
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
A094730
Number of connected ordered 3-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 25, 337, 4321, 46681, 437305, 3721537, 29740561, 228000361, 1699890985, 12435686737, 89792976001, 642488104441, 4567920215065, 32331017955937, 228106608326641, 1605738151030921, 11285298643841545, 79223419486529137
Offset: 0
-
With[{nmax = 50}, CoefficientList[Series[Exp[7*x] - 6*Exp[5*x] + 3*Exp[4*x] + 17*Exp[3*x] - 30*Exp[2*x] + 21*Exp[x] - 6, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
-
x='x+O('x^50); concat([0], Vec(serlaplace(exp(7*x) - 6*exp(5*x) + 3*exp(4*x) + 17*exp(3*x) - 30*exp(2*x) + 21*exp(x) - 6))) \\ G. C. Greubel, Oct 08 2017
A094731
Number of connected ordered 4-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 79, 2101, 63991, 1841461, 45677479, 986583781, 19210969591, 347527345621, 5968468471879, 98788140462661, 1592387628858391, 25181074712937781, 392680081411090279, 6061279724768728741, 92859536016650958391, 1414764491802643937941
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..845
- Index entries for linear recurrences with constant coefficients, signature (63,-1701,25887,-245427,1510257,-6084119,15754053,-24891552,21416940,-7484400).
-
With[{nmax = 50}, CoefficientList[Series[Exp[15*x] - 12*Exp[11*x] + 24*Exp[9*x] - 8*Exp[7*x] + 27*Exp[6*x] - 96*Exp[5*x] - 6*Exp[4*x] + 246*Exp[3*x] - 288*Exp[2*x] + 138*Exp[x] - 26, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
-
x='x+O('x^50); concat([0], Vec(serlaplace(exp(15*x) -12*exp(11*x) +24*exp(9*x) -8*exp(7*x) +27*exp(6*x) -96*exp(5*x) -6*exp(4*x) +246*exp(3*x) -288*exp(2*x) +138*exp(x) -26))) \\ G. C. Greubel, Oct 08 2017
-
concat(0, Vec(x*(1 - 62*x + 1717*x^2 - 27062*x^3 + 285547*x^4 - 1926074*x^5 + 8088135*x^6 - 28645362*x^7 + 105534360*x^8 - 194594400*x^9) / ((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
A094732
Number of connected ordered 5-element multiantichains on a labeled n-set.
Original entry on oeis.org
0, 1, 1, 241, 11761, 736801, 50524321, 3176975761, 171220124881, 8021076673921, 337296669440641, 13098877345981681, 479949442942292401, 16851170646696553441, 573314381587074123361, 19054886956855687698001
Offset: 0
-
With[{nmax = 50}, CoefficientList[Series[Exp[31*x] - 20*Exp[23*x] + 60*Exp[19*x] + 20*Exp[17*x] + 5*Exp[16*x] - 95*Exp[15*x] - 120*Exp[14*x] + 150*Exp[13*x] + 180*Exp[12*x] - 420*Exp[11*x] - 110*Exp[10*x] + 620*Exp[9*x] + 160*Exp[8*x] - 690*Exp[7*x] + 840*Exp[6*x] - 936*Exp[5*x] - 1140*Exp[4*x] + 3560*Exp[3*x] - 3010*Exp[2*x] + 1095*Exp[x] - 150, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 08 2017 *)
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