I will show how the $R$-transform of free probability can be used in order to obtain bounds on the asymptotic value of $E(X_n)/n$, where $X_n$ is the random variable counting the components of a random meandric system on $2n$ points. This follows a joint work with Ian Goulden and Doron Puder (in IMRN 2020(4), pp. 983-1034).