Éric Fusy has authored 4 sequences.
A363682
Number of plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) with n edges.
Original entry on oeis.org
1, 1, 1, 2, 5, 12, 32, 93, 279, 872, 2830, 9433, 32223, 112527, 400370, 1448520, 5320023, 19802827, 74612164, 284238390, 1093757436, 4247742956, 16636921148, 65671960544, 261111950308, 1045172796381, 4209807155949, 17055625810984, 69476952146529, 284467866640048
Offset: 1
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A:=proc(n,i,j) option remember:
if n=0 and i=0 and j=0 then return 1:
elif n<=0 or j<0 or i<0 then return 0:
else
return A(n-1,i,j)+A(n-1,i-1,j)+A(n-1,i,j+1)+A(n-1,i+1,j-1)+A(n-1,i-1,j+1):
fi:
end proc:
seq(A(n-1,0,0),n=1..20);
A114289
Number of combinatorial types of n-dimensional polytopes with n+3 vertices.
Original entry on oeis.org
0, 1, 7, 31, 116, 379, 1133, 3210, 8803, 23701, 63239, 168287, 447905, 1194814, 3196180, 8576505, 23081668, 62292381, 168536249, 457035453, 1241954405, 3381289332, 9221603416, 25189382006, 68906572413, 188750887991
Offset: 1
Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005
- B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
- Lukas Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001. See Table 7.5.
- Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.
- Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
- E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.
- Aleksandr Maksimenko, 2-neighborly 0/1-polytopes of dimension 7, arXiv:1904.03638 [math.CO], 2019.
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N:=60: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-ln(1-2*x)+ln(1-x): K:=-1/2*x*(x-8*x^3-1+5*x^2-7*x^4+2*x^6+5*x^8-9*x^7+19*x^5-14*x^9+x^10+19*x^11-5*x^12+4*x^14-8*x^13)/(1-x)^5/(2*x^6-4*x^4+4*x^2-1)/(x+1)^2: series(1/(x^3-x^4)*(1/4*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+1/2*sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);
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terms = 26;
G[x_] = -Log[1 - 2(x^3/(1 - 2x)^2)];
H[x_] = -Log[1 - 2x] + Log[1 - x];
K[x_] = -1/2 x (x - 8x^3 - 1 + 5x^2 - 7x^4 + 2x^6 + 5x^8 - 9x^7 + 19x^5 - 14x^9 + x^10 + 19x^11 - 5x^12 + 4x^14 - 8x^13)/(1-x)^5/(2x^6 - 4x^4 + 4x^2 - 1)/(x+1)^2;
1/(x^3 - x^4) (1/4 Sum[EulerPhi[2r + 1]/(2r + 1) G[x^(2r + 1)], {r, 0, terms+2}] + 1/2 Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+2}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* Jean-François Alcover, Dec 14 2018 *)
A114290
Number of oriented n-dimensional polytopes with n+3 vertices, meaning that two polytopes are identified if they have the same combinatorial type and there exists an orientation-preserving homeomorphism mapping the first polytope to the second polytope.
Original entry on oeis.org
0, 1, 7, 38, 170, 617, 1979, 5859, 16571, 45516, 123159, 330736, 885780, 2372305, 6362965, 17102719, 46078541, 124440388, 336829857, 913658780, 2483217288, 6761405513, 18441239903, 50375429081, 137807555515, 377492301876
Offset: 1
Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005
- B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
- Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.
- Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
- E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.
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N:=30: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-log(1-2*x)+ln(1-x): K:=-(x^10+3*x^9-3*x^8-7*x^7+4*x^6+4*x^5+4*x^4+3*x^3-2*x^2+1)*x/(1-x)^5/(x+1)^3: series(1/(x^3-x^4)*(1/2*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);
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terms = 26;
G[x_] = -Log[1 - 2 (x^3/(1 - 2 x)^2)];
H[x_] = -Log[1 - 2 x] + Log[1 - x];
K[x_] = -(x^10 + 3 x^9 - 3 x^8 - 7 x^7 + 4 x^6 + 4 x^5 + 4 x^4 + 3 x^3 - 2 x^2 + 1) x/(1 - x)^5/(x + 1)^3;
1/(x^3 - x^4) (1/2 Sum[EulerPhi[2 r + 1]/(2 r + 1) G[x^(2 r + 1)], {r, 0, terms+3}] + Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+3}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* Jean-François Alcover, Dec 14 2018 *)
A114291
Number of combinatorial types of achiral n-dimensional polytopes with n+3 vertices, where a polytope is achiral if one of its geometric realizations has a reflection-symmetry.
Original entry on oeis.org
0, 1, 7, 24, 62, 141, 287, 561, 1035, 1886, 3319, 5838, 10030, 17323, 29395, 50291, 84795, 144374, 242641, 412126, 691522, 1173151, 1966929, 3334931, 5589311, 9474106, 15875699, 26906538, 45083426, 76404103, 128014623, 216944163
Offset: 1
Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005
- B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
- Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.
- Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
- E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.
- Index entries for linear recurrences with constant coefficients, signature (2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2).
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LinearRecurrence[{2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2}, {0, 1, 7, 24, 62, 141, 287, 561, 1035, 1886, 3319, 5838, 10030, 17323}, 32] (* Jean-François Alcover, Dec 14 2018 *)
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concat(0, Vec((2*x^11+4*x^10-2*x^9-15*x^8-5*x^7+23*x^6+15*x^5 -17*x^4-14*x^3+4*x^2 +5*x+1)*x^2/ (-1+x)^5/(2*x^6-4*x^4+4*x^2-1)/(x+1)^3 + O(x^50))) \\ Michel Marcus, Dec 12 2014
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