Notes that Tom took during meeting with Sara on Friday:

list of isomorphism classes of subintervals hasse diagram partial interval k-crowns: ranked degree sequences (and up vs. down) Ekedahl (Ekedahl structure interval) Kazhdan - Lusztig polynomial Questions: when is [id,v] ~ [id,w] ? completely determined by heights & degrees? count # interval isomorphism classes 1) [id,w] 2) all Pattern avoidance: (general idea) for n = 1,2,3... for all w \in Sn if (has_property(w) == avoids_pattern(w)): 'good' else print has_property(w), avoids_patterns(w) add w to known patterns (specific examples) Smoothness: w is a smooth permutation iff #{Tij <= w} = h(w) (i < j) 3412 or 4231 avoiding Fully commutative perms: w avoids 321