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Bruce Sagan
Bruce Sagan
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TheOriginally I thought that the answer to your question is yes. This cancould 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 interiora given set of peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using Unfortunately, the Theorem 1.1theorem only applies if the number of elements in the cited paper shows that a_npeak set is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}constant with respect to n, the ratio clearly goeswhich is not true in this case. 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.

The answer to your question is yes. This can be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly the number of permutations you are considering is less than the number, a_n, which have interior peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using the Theorem 1.1 in the cited paper shows that a_n is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes to zero.

Cheers, Bruce Sagan

PS Thanks to Richard Stanley for pointing out this post to us.

Originally I thought that the answer to your question could be deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Theorem 1.1 in that paper gives an enumeration result for the number of permutations in S_n which have a given set of peaks (what you are calling interior local maxima). Unfortunately, the theorem only applies if the number of elements in the peak set is constant with respect to n, which is not true in this case. It would be very interesting to find 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.

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Bruce Sagan
Bruce Sagan
  • 101
  • 1
  • 3

The answer to your question is yes. This can be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly the number of permutations you are considering is less than the number, a_n, which have interior peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using the Theorem 1.1 in the cited paper shows that a_n is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes to zero.

Cheers, Bruce Sagan

PS Thanks to Richard Stanley for pointing out this post to us.

The answer to your question is yes. This can be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly the number of permutations you are considering is less than the number, a_n, which have interior peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using the Theorem 1.1 in the cited paper shows that a_n is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes to zero.

Cheers, Bruce Sagan

The answer to your question is yes. This can be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly the number of permutations you are considering is less than the number, a_n, which have interior peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using the Theorem 1.1 in the cited paper shows that a_n is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes to zero.

Cheers, Bruce Sagan

PS Thanks to Richard Stanley for pointing out this post to us.

Source Link
Bruce Sagan
Bruce Sagan
  • 101
  • 1
  • 3

The answer to your question is yes. This can be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly the number of permutations you are considering is less than the number, a_n, which have interior peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using the Theorem 1.1 in the cited paper shows that a_n is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes to zero.

Cheers, Bruce Sagan