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A357593 Number of faces of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector.

%I A357593 #10 Nov 19 2022 20:28:44
%S A357593 8,26,88,298,1016,3466,11832,40394,137912,470858
%N A357593 Number of faces of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector.
%H A357593 L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, <a href="https://arxiv.org/abs/2209.14978">Enumeration of max-pooling responses with generalized permutohedra</a>, arXiv:2209.14978 [math.CO], 2022. (See Table 3)
%e A357593 For n=1, the polytope is the simplex with vertices (1,0,0), (0,1,0), and (0,0,1) that has a(1)=8 faces (1 empty face, 3 vertices, 3 edges, and 1 facet).
%o A357593 (Sage) def a(n): return add(PP(n,3,1).f_vector())
%o A357593 def Delta(I,n):
%o A357593     IM = identity_matrix(n)
%o A357593     return Polyhedron(vertices=[IM[e] for e in I],backend='normaliz')
%o A357593 def Py(n,SL,yL):
%o A357593     return sum(yL[i]*Delta(SL[i],n) for i in range(len(SL)))
%o A357593 def PP(n,k,s):
%o A357593     SS = [set(range(s*i,k+s*i)) for i in range(n)],[1,]*(n)
%o A357593     return Py(s*(n-1)+k,SS[0],SS[1])
%o A357593 [a(n) for n in range(1,4)]
%Y A357593 Cf. A033303, A007070.
%K A357593 nonn,hard,more
%O A357593 1,1
%A A357593 _Alejandro H. Morales_, Oct 05 2022