The
Originally I thought that the answer to your question is yes. This can could be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly Theorem 1.1 in that paper gives an enumeration result for the number of permutations you are considering is less than the number, a_n, in S_n which have interior a given set of peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywheremaxima). A simple computation using Unfortunately, the Theorem 1.1 theorem only applies if the number of elements in the cited paper shows that a_n peak set is at most p(n) 2^{2n} for some polynomial p(n)constant with respect to n, which is not true in this case. But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes It would be very interesting to zerofind an analogue of this theorem where the size of the peak set varies with n.
Cheers, Bruce Sagan
PS Thanks to Richard Stanley for pointing out this post to us.
