  We show that the polytopes obtained from the Birkhoff polytope by imposing
additional inequalities restricting the "longest increasing subsequence" have
Ehrhart quasi-polynomials which are honest polynomials, even though they are
just rational polytopes in general. We do this by defining a continuous,
piecewise-linear bijection to a certain Gelfand-Tsetlin polytope. This
bijection is not an integral equivalence but it respects lattice points in the
appropriate way to imply that the two polytopes have the same Ehrhart
(quasi-)polynomials. In fact, the bijection is essentially the
Robinson-Schensted-Knuth correspondence.
