  Given two vectors $u$ and $v$, their outer sum is given by the matrix $A$
with entries $A_{ij} = u_{i} + v_{j}$. If the entries of $u$ and $v$ are
increasing and sufficiently generic, the total ordering of the entries of the
matrix is a standard Young tableau of rectangular shape. We call standard Young
tableaux arising in this way realizable. The set of realizable tableaux was
defined by Mallows and Vanderbei for studying a deconvolution algorithm, but we
show they have appeared in many other contexts including sorting algorithms,
quantum computing, random sorting networks, reflection arrangements, fiber
polytopes, and Goodman and Pollack's theory of allowable sequences. In our
work, we prove tight bounds on the asymptotic number of realizable rectangular
tableaux. We also derive tight asymptotics for the number of realizable
allowable sequences, which are in bijection with realizable staircase-shaped
standard Young tableaux with the notion of realizability coming from the theory
of sorting networks. As a consequence, we resolve an open question of Angel,
Gorin, and Holroyd from 2012 and improve upon a 1986 result of Goodman and
Pollack.
