We study stable matchings between \(n\) men and \(n\) women. Each individual strictly ranks \(n\) members of the other group. Each individual's preferences are independently drawn from the Mallows distribution with parameter \(q\in (0,1)\). The rankings favor a shared reference order for each group. For every fixed \(q\), we show that the logarithm of the number of stable matchings divided by \(n\), converges in probability to a finite positive growth rate. And this growth rate is continuous in \(q\) and tends to zero as \(q\) approaches zero. Furthermore we show that after centering the logarithm of the number of stable matchings by \(n\) times the growth rate and dividing by \(\sqrt n\), the limiting distribution is normal with finite positive variance.