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Under the Flaschka-Newell Lax pair, the Darboux transformation for thePainlev\'{e}-II equation is constructed by the limiting technique. With the aidof the Darboux transformation, the rational solutions are represented by theGram determinant, and then we give the large $y$ asymptotics of the determinantand the rational solutions. Finally, the solution of the correspondingRiemann-Hilbert problem is obtained from the Darboux matrices.

Rational solutions of Painlev\'{e}-II equation as Gram determinant