The sum of width-one tensors

Abstract

: This paper generalizes a recent result concerning the sum of width-one matrices; in the present work, we consider width-one tensors of arbitrary dimensions. A tensor is said to be width-one if, when visualized as an array, its nonzero entries lie along a path consisting of steps in the positive directions of the standard coordinate vectors. We prove two 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 (thereby inverting the northwest corner rule from optimal transport theory). 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.