  This paper generalizes a recent result by the authors concerning the sum of
width-one matrices; in the present work, we consider width-one tensors of
arbitrary dimensions. A tensor is said to have width 1 if, when visualized as
an array, its nonzero entries lie along a path consisting of steps in the
directions of the standard coordinate vectors. We prove two different formulas
to compute the sum of all width-one tensors with fixed dimensions and fixed sum
of (nonnegative integer) components. The first formula is obtained by
converting width-one tensors into tuples of one-row semistandard Young
tableaux; the second formula, which extracts coefficients from products of
multiset Eulerian polynomials, is derived via Stanley-Reisner theory, making
use of the EL-shelling of the order complex on the standard basis of tensors.
