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:$$1 \;\; 3$$
$$2\;\;\;$$