  The weighted projection of an alternating sign matrix (ASM) was introduced by
Brualdi and Dahl (2018) as a step towards characterising a generalisation of
Latin squares they introduced using alternating sign hypermatrices. If $z_n =
(n,\dots,2,1)$, then the weighted projection of an ASM $A$ is equal to
$z_n^TA$. Brualdi and Dahl proved that the weighted projection of an $n \times
n$ ASM is majorized by the vector $z_n$, and conjectured that any positive
integer vector majorized by $z_n$ is the weighted projection of some ASM. The
main result of this paper presents a proof of this conjecture, via monotone
triangles. A relaxation of a monotone triangle, called a row-increasing
triangle, is introduced. It is shown that for any row-increasing triangle $T$,
there exists a monotone triangle $M$ such that each entry of $M$ occurs the
same number of times as in $T$. A construction is also outlined for an ASM with
given weighted projection. The relationship of the main result to existing
results concerning the ASM polytope $ASM_n$ is examined, and a characterisation
is given for the relationship between elements of $ASM_n$ corresponding to the
same point in the permutohedron of order $n$. Finally, the limitations of the
main result for characterising alternating sign hypermatrix Latin-like squares
is considered.
