A307389 a(n) is the number of elements in the species of orbit polytopes in dimension n.
1, 1, 2, 7, 29, 136, 737, 4537, 30914, 229831, 1850717, 16036912, 148573889, 1463520241, 15259826402, 167789512807, 1939125333629, 23484982837816, 297289975208417, 3924325664733097, 53906145745657634, 769095929901073831, 11377500925452103037, 174244037885068510432
Offset: 0
Keywords
Examples
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}.
Links
- Jinyuan Wang, Table of n, a(n) for n = 0..517
- Mariel Supina, The Hopf Monoid of Orbit Polytopes, arXiv:1904.08437 [math.CO], 2019.
Crossrefs
Cf. A376544.
Programs
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Magma
m:=30; R
:=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 -
Maple
b:= proc(n, p) option remember; `if`(n=0, 1/p!, add( b(n-j, 0)*binomial(n-1, j-1)/p!, j=2..n)+b(n-1, p+1)*n) end: a:= n-> b(n, 0): seq(a(n), n=0..23); # Alois P. Heinz, Dec 01 2024 -
Mathematica
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 *) -
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]
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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
Formula
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]
Extensions
More terms from Michel Marcus, Apr 26 2019
Comments