Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .

A permutation is called a *Roller Coaster permutation* if . Let be the set of all Roller Coaster permutations in .

**Conjecture**For ,

- \item If , then . \item If , then with .

**Conjecture (Odd Sum conjecture)**Given ,

- \item If , then is odd for . \item If , then for all .

## Bibliography

*[AS] Tanbir Ahmed, Hunter Snevily, Some properties of Roller Coaster permutations. To appear in *Bull. Institute of Combinatorics and its Applications*, 2013.

* indicates original appearance(s) of problem.