0,3

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.

Jinyuan Wang, Table of n, a(n) for n = 0..517

Ira Gessel, Recursion for the A307389, answer to question on MathOverflow, 2024.

Eddie Nijholt, Nándor Sieben, and James W. Swift, Invariant Synchrony and Anti-Synchrony Subspaces of Weighted Networks, arXiv:2206.00094 [math.DS], 2022-2023. See pp. 24, 28.

Mariel Supina, The Hopf Monoid of Orbit Polytopes, arXiv:1904.08437 [math.CO], 2019.

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.

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}.

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

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

(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

(SageMath) 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

Cf. A376544.

Sequence in context: A074600 A064641 A183608 * A104252 A391302 A373802

Adjacent sequences: A307386 A307387 A307388 * A307390 A307391 A307392

Mariel Supina, Apr 17 2019

More terms from Michel Marcus, Apr 26 2019

approved