An extended generalization of RSK correspondence via $A$ type quiver representations

Let $\lambda =({\lambda }_1⩾\dots ⩾{\lambda }_k>0)$. For any $c$ Coxeter element of ${\mathfrak{S}}_{{\lambda }_1+k-1}$, we construct a bijection from fillings of $\lambda$ to reverse plane partitions. We recover two previous generalizations of the Robinson–Schensted–Knuth correspondence for particular choices of Coxeter element depending on $\lambda$: one based on the work of, among others, Burge, Hillman, Grassl, Knuth, and uniformly presented by Gansner; the other developed by Garver, Partrias, and Thomas, and independently by Dauvergne, called Scrambled RSK. Our results in this paper develop the combinatorial consequence of our previous work of type $A_{{\lambda }_1+k}$ quivers.