  Associated to Legendrian links in the standard contact three-space, Ruling
polynomials are Legendrian isotopy invariants, which also compute augmentation
numbers, that is, the points-counting of augmentation varieties for Legendrian
links (up to a normalized factor) \cite{HR15}. In this article, we generalize
this picture to Legendrian tangles, which are morally the pieces obtained by
cutting Legendrian link fronts along 2 vertical lines. Moreover, we show that
the Ruling polynomials for Legendrian tangles satisfy the composition axiom. In
the special case of Legendrian knots, our arguments provide new proofs to the
main results in \cite{HR15}. In the end, we also introduce generalized Ruling
polynomials for Legendrian tangles, to account for non-acyclic augmentations in
the "Ruling polynomials compute augmentation numbers" picture.
