Four transformations on the Catalan triangle

In this paper, we define four transformations on the classical Catalan triangle $\mathcal{C}=(C_{n,k})_{n\geq k\geq 0}$ with $C_{n,k}=\frac{k+1}{n+1}\binom{2n-k}{n}$. The first three ones are based on the determinant and the forth is utilizing the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combinatorial triangles.