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

Restricted Birkhoff polytopes and Ehrhart period collapse