Combinatorial interpretation of Kazhdan–Lusztig basis elements indexed by 45312-avoiding permutations in 𝔖₆

Deodhar [Geom. Dedicata 36, no. 1 (1990)] introduced the defect statistic on subexpressions of reduced expressions in the symmetric group 𝔖_n to construct an algorithmic description of the Kazhdan–Lusztig basis of the Hecke algebra H_n(q). This led Billey–Warrington [J. Algebraic Combin. 13, no. 2 (2001)] and the second author [J. Pure Appl. Algebra 212 (2008)] to state very explicit combinatorial descriptions of the basis elements indexed by permutations avoiding certain patterns. We extend the above work by producing an exhaustive list of graphical representations of Kazhdan–Lusztig basis elements indexed by 45312-avoiding permutations w ∈𝔖₅, 𝔖₆.