The KL right cell is the collection of permutations that have the same insertion tableau S of by RSK algorithm, for the left cell is the recording tableau T. And KL-right cell is the equivalence class of KL preorder on $S_n$.

The S of 312 is : $$1 \;\; 2$$ $$3\;\;\;$$ The T is:$$1 \;\; 3$$ $$2\;\;\;$$

The KL right cell is the collection of permutations that have the same insertion tableau S from RSK algorithm, the left cell is those of the same recording tableau T. KL right(left) cell are the equivalence classes of KL right(left) preorder on $\mathfrak{S}_n$.

The S of 312 is : $$1 \;\; 2$$ $$3\;\;\;$$ The T is:$$1 \;\; 3$$ $$2\;\;\;$$