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A307389
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a(n) is the number of elements in the species of orbit polytopes in dimension n.
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1
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1, 1, 2, 7, 29, 136, 737, 4537, 30914, 229831, 1850717, 16036912, 148573889, 1463520241, 15259826402, 167789512807, 1939125333629, 23484982837816, 297289975208417, 3924325664733097, 53906145745657634, 769095929901073831, 11377500925452103037, 174244037885068510432
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OFFSET
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0,3
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COMMENTS
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An orbit polytope is a polytope whose vertices are all of the permutations of the coordinates of some point in R^n. Two polytopes are normally equivalent if they have the same normal fan. The species of orbit polytopes maps a finite set I to the set OP[I] of normal equivalence classes of finite products of orbit polytopes in RI. For each n, this sequence counts the size of OP[I] when |I|=n.
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LINKS
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FORMULA
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E.g.f.: exp((exp(2*t) - 2*exp(t) + 2*t + 1)/2). This is because OP is the exponential of the species of compositions mapping a finite set I to the set of compositions of the integer |I|, excluding compositions with one part if |I|>1.
a(n) = R(n, 0) for n >= 0 where R(n, q) = (q+1)*R(n-1, q) - R(n-1, q+1) + R(n-1, q+2) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Jan 04 2024 [verification needed]
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EXAMPLE
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For n=3, there are 7 normal equivalence classes. Among these are the 4 normal equivalence classes of orbit polytopes in R^3: the permutohedron conv{123,132,213,231,321,312}, the standard simplex conv{100,010,001}, the simplex conv{110,101,011}, and a point. In addition, there are 3 normal equivalence classes of products of two orbit polytopes, which are the line segments conv{001,010}, conv{001,100}, and conv{010,100}.
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MATHEMATICA
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nmax = 30; CoefficientList[Series[E^((E^(2*x) - 2*E^x + 2*x + 1)/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 18 2019 *)
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PROG
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(PARI) my(t='t+O('t^30)); Vec(serlaplace(exp((exp(2*t)-2*exp(t)+2*t+1 )/2))) \\ Michel Marcus, Apr 24 2019
(PARI) upto(n) = my(v1, v2, v3); v1 = vector(2*n + 1, i, 1); v2 = v1; v3 = vector(n + 1, i, 0); v3[1] = 1; for(i = 1, n, for(q = 0, 2*(n - i), v2[q + 1] = (q + 1) * v1[q + 1] - v1[q + 2] + v1[q + 3]); v1 = v2; v3[i + 1] = v1[1]); v3 \\ Mikhail Kurkov, Jan 04 2024 [verification needed]
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp((Exp(2*x) -2*Exp(x) +2*x +1)/2) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 14 2019
(Sage) m = 30; T = taylor(exp((exp(2*x) -2*exp(x) +2*x +1)/2), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jul 14 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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