This  question  has its origin in combinatorial topology. In the  90s R. Forman proposed a  discrete counterpart of Morse theory. In   his  case, a Morse function on a triangulated space is a function that assigns   a number to each  face and satisfying certain  conditions.

The theory has found   beautiful applications, but  it has one limitation: discrete  Morse functions   are hard to find, unlike the smooth case where  smooth Morse functions are a dime a dozen.  That made me think that maybe  that aren't  too many such discrete  Morse functions.     So naturally one can ask, really, how many Morse functions are out there on a given triangulated space. 

The present question  deals with the  simplest triangulated space, namely a line segment divided into $n$-subintervals. The probelm of counting the combinatorial Morse functions on this triangulated space reduces to  the following  purely combinatorial problem.


Consider the group $S_{2n+1}$ of permutations of the set 

$$ V_n:=\lbrace 0,1,\dotsc,2n\rbrace. $$

 A point $i\in V_n$ is called an *interior point*  if $i\neq 0,2n$. An interior point $i\in V_n$ is a *local minimum*  of a permutation $\phi\in S_{2n+1}$ if

$$ \phi(i-1)> \phi(i) <\phi(i+1). $$

A *local maximum* is defined  in a similar fashion.   Here is now the question.


> Denote  by $p_n$ the probability that
> a random permutation of $S_{2n+1}$ has
> the property that all its **interior**
> local minima (**if any**) are even and all its
> **interior** local maxima (**if any**) are odd. Is it
> true (as I believe) that
> $p_n\to 0$ as $n\to\infty$? Can
> one   be more precise  about the
> behavior of $p_n$ as $n\to\infty$? 


For example, when $n=1$ the permutations  of $\lbrace0,1,2\rbrace$ satisfying the above constraints are

$$ (0,1,2), (2,1,0), (0,2,1), (1,2,0). $$

Hence $p_1=\frac{2}{3}$.

Thanks.