We introduce a weight-dependent extension of the classical Mahonian statistic on permutations, the inversion number. This immediately gives us a new weight-dependent extension of $n!$. When we restrict to $312$-avoiding permutations, our extension gives rise to a weight-dependent family of Catalan numbers, which happen to coincide with the weighted Catalan numbers that were previously introduced by Postnikov and Sagan by weighted enumeration of Dyck paths. While Postnikov and Sagan's main focus was on the modulo $2$ divisibility of the weighted Catalan numbers, we discovered further properties of these numbers that highly resemble those of the classical case, such as their recurrence relation and Hankel determinants. We will also present certain bi-weighted Catalan numbers that generalize Garsia and Haiman's $q,t$-Catalan numbers and again satisfy interesting properties. This talk is based on recent joint work with Michael Schlosser.