Importance: High ✭✭✭
 Author(s): Ahmed, Tanbir Snevily, Hunter S.
 Subject: Combinatorics
 Keywords:
 Posted by: Tanbir Ahmed on: October 14th, 2013

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.