1 IntroductionRigged configurations appear in the Bethe Ansatz study of exactly solvable lattice models ascombinatorial objects to index the solutions of the Bethe equations [5, 6]. Based on work byKerov, Kirillov and Reshetikhin [5, 6], it was shown in that there is a statistic preserving bi-jection Φ between Littlewood-Richardson tableaux and rigged configurations. The descriptionof the bijection Φ is based on a quite technical recursive algorithm.Littlewood-Richardson tableaux can be viewed as highest weight crystal elements in a ten-sor product of Kirillov–Reshetikhin (KR) crystals of type A(1)n . KR crystals are affine finite-dimensional crystals corresponding to affine Kac–Moody algebras, in the setting of of typeA(1)n . The highest weight condition is with respect to the finite subalgebra An. The bijectionΦ can be generalized by dropping the highest weight requirement on the elements in the KRcrystals , yielding the set of crystal paths P . On the corresponding set of unrestr...