so given a total order on the interval [1,n] of natural numbers, consider the collection of all subintervals for which the order restricted to them is completely backwards of the usual order...

i guess only bother with subintervals with distinct endpoints...

for example n=3:

1 123 []
2 213 [12]
2 132 [23]
3 231 [12]
3 312 [23]
4 321 [13 12 23]

ok that's not quite doing what i wanted it to... but maybe i see a way to modify it so that it will ... ??...