![](/files/happy5.png)
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
,
![$ n\geq 3 $](/files/tex/faa360f8c4c583b1d342c73b21addf9c70b4dd2e.png)
- \item If
![$ n=2k $](/files/tex/b26ae48a38c6f453cd224b1153a91d12f2e63ba2.png)
![$ |RC(n)|=4 $](/files/tex/4b8b5bc85250888a05476ac8a85130c7f2aec30f.png)
![$ n=2k+1 $](/files/tex/e93281c0bb1f46afe416bbf51dc3f4c1fdf39e3e.png)
![$ |RC(n)|=2^j $](/files/tex/d41dcf721659048454573af9e68b2bb2284d5acb.png)
![$ j\leq k+1 $](/files/tex/578b553aab9fcec9e75de29ab0b3536d16869877.png)
Conjecture (Odd Sum conjecture) Given
,
![$ \pi\in RC(n) $](/files/tex/2e85bc17b36d00ad4dd5807ec16ce84a62cdb109.png)
- \item If
![$ n=2k+1 $](/files/tex/e93281c0bb1f46afe416bbf51dc3f4c1fdf39e3e.png)
![$ \pi_j+\pi_{n-j+1} $](/files/tex/41c60f8483171880e30871c01293b84c52f70e70.png)
![$ 1\leq j\leq k $](/files/tex/75210491d0bced81f9a329a630a194cd5ea14db2.png)
![$ n=2k $](/files/tex/b26ae48a38c6f453cd224b1153a91d12f2e63ba2.png)
![$ \pi_j + \pi_{n-j+1} = 2k+1 $](/files/tex/8eccae0ec181f943235e193019e4e99cbcd9c733.png)
![$ 1\leq j\leq k $](/files/tex/75210491d0bced81f9a329a630a194cd5ea14db2.png)
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.