Pattern avoidance and Young tableaux (Q510311)

Summary: This paper extends Lewis's bijection [\textit{J. B. Lewis}, J. Comb. Theory, Ser. A 118, No. 4, 1436--1450 (2011; Zbl 1231.05286)] 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 \[ |\mathcal{L}(n,k,I)|=f^{(k+1)^{n}\rangle}, \] 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 \(I'\).