This paper extends Lewis's bijection (J. Combin. Theorey Ser. A 118, 2011) to a bijection between a more general class $\mathcal{L}(n,k,I)$ of permutations and the set of standard Young tableaux of shape $\langle (k+1)^n\rangle$, so the cardinality\[|\mathcal{L}(n,k,I)|=f^{\langle (k+1)^n\rangle},\]is independent of the choice of $I\subseteq [n]$. As a consequence, we obtain some new combinatorial realizations and identities on Catalan numbers. In the end, we raise a problem on finding a bijection between $\mathcal{L}(n,k,I)$ and $\mathcal{L}(n,k,I')$ for distinct $I$ and $I'$.