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