  Given a partition l and a composition b, the stretched Kostka coefficient
K_{l, b}(n) is the map sending each positive integer n to the Kostka
coefficient indexed by nl and nb. Kirillov and Reshetikhin (1986) have shown
that stretched Kostka coefficients are polynomial functions of n. King, Tollu,
and Toumazet have conjectured that these polynomials always have nonnegative
coefficients (2004), and they have given a conjectural expression for their
degrees (2006).
  We prove the values conjectured by King, Tollu, and Toumazet for the degrees
of stretched Kostka coefficients. Our proof depends upon the polyhedral
geometry of Gelfand--Tsetlin polytopes and uses tilings of GT-patterns, a
combinatorial structure introduced in De Loera & McAllister (2004).
