  We establish relationships between two classes of invariants of Legendrian
knots in $\mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA
and satellite ruling polynomials. For positive permutation braids, $\beta
\subset J^1S^1$, we give a precise formula in terms of representation numbers
for the $m$-graded ruling polynomial $R^m_{S(K,\beta)}(z)$ of the satellite of
$K$ with $\beta$ specialized at $z=q^{1/2}-q^{-1/2}$ with $q$ a prime power,
and we use this formula to prove that arbitrary $m$-graded satellite ruling
polynomials, $R^m_{S(K,L)}$, are determined by the Chekanov-Eliashberg DGA of
$K$. Conversely, for $m\neq 1$, we introduce an $n$-colored $m$-graded ruling
polynomial, $R^m_{n,K}(q)$, in strict analogy with the $n$-colored HOMFLY-PT
polynomial, and show that the total $n$-dimensional $m$-graded representation
number of $K$ to $\mathbb{F}_q^n$, $\mbox{Rep}_m(K,\mathbb{F}_q^n)$, is exactly
equal to $R^m_{n,K}(q)$. In the case of $2$-graded representations, we show
that $R^2_{n,K}=\mbox{Rep}_2(K, \mathbb{F}_q^n)$ arises as a specialization of
the $n$-colored HOMFLY-PT polynomial.
