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.

8, 26, 88, 298, 1016, 3466, 11832, 40394, 137912, 470858

Offset

1,1

Links

Table of n, a(n) for n=1..10.

L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022. (See Table 3)

Example

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).

Prog

(Sage) def a(n): return add(PP(n, 3, 1).f_vector())
def Delta(I, n):
    IM = identity_matrix(n)
    return Polyhedron(vertices=[IM[e] for e in I], backend='normaliz')
def Py(n, SL, yL):
    return sum(yL[i]*Delta(SL[i], n) for i in range(len(SL)))
def PP(n, k, s):
    SS = [set(range(s*i, k+s*i)) for i in range(n)], [1, ]*(n)
    return Py(s*(n-1)+k, SS[0], SS[1])
[a(n) for n in range(1, 4)]

Crossrefs

Cf. A033303, A007070.

Sequence in context: A194021 A245126 A278769 * A173365 A261971 A140788

Adjacent sequences: A357590 A357591 A357592 * A357594 A357595 A357596

Keyword

nonn,hard,more

Author

Alejandro H. Morales, Oct 05 2022

Status

approved