cp's OEIS Frontend

A307389 a(n) is the number of elements in the species of orbit polytopes in dimension n.

%I A307389 #42 Dec 01 2024 20:40:31
%S A307389 1,1,2,7,29,136,737,4537,30914,229831,1850717,16036912,148573889,
%T A307389 1463520241,15259826402,167789512807,1939125333629,23484982837816,
%U A307389 297289975208417,3924325664733097,53906145745657634,769095929901073831,11377500925452103037,174244037885068510432
%N A307389 a(n) is the number of elements in the species of orbit polytopes in dimension n.
%C A307389 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.
%H A307389 Jinyuan Wang, <a href="/A307389/b307389.txt">Table of n, a(n) for n = 0..517</a>
%H A307389 Mariel Supina, <a href="https://arxiv.org/abs/1904.08437">The Hopf Monoid of Orbit Polytopes</a>, arXiv:1904.08437 [math.CO], 2019.
%F A307389 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.
%F A307389 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]
%e A307389 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}.
%p A307389 b:= proc(n, p) option remember; `if`(n=0, 1/p!, add(
%p A307389       b(n-j, 0)*binomial(n-1, j-1)/p!, j=2..n)+b(n-1, p+1)*n)
%p A307389     end:
%p A307389 a:= n-> b(n, 0):
%p A307389 seq(a(n), n=0..23);  # _Alois P. Heinz_, Dec 01 2024
%t A307389 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 *)
%o A307389 (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
%o A307389 (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]
%o A307389 (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
%o A307389 (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
%Y A307389 Cf. A376544.
%K A307389 nonn
%O A307389 0,3
%A A307389 _Mariel Supina_, Apr 17 2019
%E A307389 More terms from _Michel Marcus_, Apr 26 2019