Castelnuovo-Mumford regularity for $321$-avoiding Kazhdan-Lusztig varieties

We prove the Castelnuovo–Mumford regularity of 321-avoiding Kazhdan–Lusztig varieties can be computed combinatorially in terms of $K$-theoretic skew excited Young diagrams. We present an algorithm which gives a lower bound for this regularity and describe a setting in which this algorithm provides precise regularity computations. This algorithm specializes to compute the regularity of all two-sided mixed ladder determinantal varieties.