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Statistic identifier: St001948

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Collection: Permutations

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Description: The number of augmented double ascents of a permutation.

An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.

A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.

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References: 

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Code:
def statistic(pi):
    pi = [0] + list(pi)
    return sum(1 for i in range(1,len(pi)-1) if pi[i-1] < pi[i] < pi[i+1])

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Statistic values:

[1,2]       => 1
[2,1]       => 0
[1,2,3]     => 2
[1,3,2]     => 1
[2,1,3]     => 0
[2,3,1]     => 1
[3,1,2]     => 0
[3,2,1]     => 0
[1,2,3,4]   => 3
[1,2,4,3]   => 2
[1,3,2,4]   => 1
[1,3,4,2]   => 2
[1,4,2,3]   => 1
[1,4,3,2]   => 1
[2,1,3,4]   => 1
[2,1,4,3]   => 0
[2,3,1,4]   => 1
[2,3,4,1]   => 2
[2,4,1,3]   => 1
[2,4,3,1]   => 1
[3,1,2,4]   => 1
[3,1,4,2]   => 0
[3,2,1,4]   => 0
[3,2,4,1]   => 0
[3,4,1,2]   => 1
[3,4,2,1]   => 1
[4,1,2,3]   => 1
[4,1,3,2]   => 0
[4,2,1,3]   => 0
[4,2,3,1]   => 0
[4,3,1,2]   => 0
[4,3,2,1]   => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 3
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
[1,5,2,3,4] => 2
[1,5,2,4,3] => 1
[1,5,3,2,4] => 1
[1,5,3,4,2] => 1
[1,5,4,2,3] => 1
[1,5,4,3,2] => 1
[2,1,3,4,5] => 2
[2,1,3,5,4] => 1
[2,1,4,3,5] => 0
[2,1,4,5,3] => 1
[2,1,5,3,4] => 0
[2,1,5,4,3] => 0
[2,3,1,4,5] => 2
[2,3,1,5,4] => 1
[2,3,4,1,5] => 2
[2,3,4,5,1] => 3
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[2,4,1,3,5] => 2
[2,4,1,5,3] => 1
[2,4,3,1,5] => 1
[2,4,3,5,1] => 1
[2,4,5,1,3] => 2
[2,4,5,3,1] => 2
[2,5,1,3,4] => 2
[2,5,1,4,3] => 1
[2,5,3,1,4] => 1
[2,5,3,4,1] => 1
[2,5,4,1,3] => 1
[2,5,4,3,1] => 1
[3,1,2,4,5] => 2
[3,1,2,5,4] => 1
[3,1,4,2,5] => 0
[3,1,4,5,2] => 1
[3,1,5,2,4] => 0
[3,1,5,4,2] => 0
[3,2,1,4,5] => 1
[3,2,1,5,4] => 0
[3,2,4,1,5] => 0
[3,2,4,5,1] => 1
[3,2,5,1,4] => 0
[3,2,5,4,1] => 0
[3,4,1,2,5] => 2
[3,4,1,5,2] => 1
[3,4,2,1,5] => 1
[3,4,2,5,1] => 1
[3,4,5,1,2] => 2
[3,4,5,2,1] => 2
[3,5,1,2,4] => 2
[3,5,1,4,2] => 1
[3,5,2,1,4] => 1
[3,5,2,4,1] => 1
[3,5,4,1,2] => 1
[3,5,4,2,1] => 1
[4,1,2,3,5] => 2
[4,1,2,5,3] => 1
[4,1,3,2,5] => 0
[4,1,3,5,2] => 1
[4,1,5,2,3] => 0
[4,1,5,3,2] => 0
[4,2,1,3,5] => 1
[4,2,1,5,3] => 0
[4,2,3,1,5] => 0
[4,2,3,5,1] => 1
[4,2,5,1,3] => 0
[4,2,5,3,1] => 0
[4,3,1,2,5] => 1
[4,3,1,5,2] => 0
[4,3,2,1,5] => 0
[4,3,2,5,1] => 0
[4,3,5,1,2] => 0
[4,3,5,2,1] => 0
[4,5,1,2,3] => 2
[4,5,1,3,2] => 1
[4,5,2,1,3] => 1
[4,5,2,3,1] => 1
[4,5,3,1,2] => 1
[4,5,3,2,1] => 1
[5,1,2,3,4] => 2
[5,1,2,4,3] => 1
[5,1,3,2,4] => 0
[5,1,3,4,2] => 1
[5,1,4,2,3] => 0
[5,1,4,3,2] => 0
[5,2,1,3,4] => 1
[5,2,1,4,3] => 0
[5,2,3,1,4] => 0
[5,2,3,4,1] => 1
[5,2,4,1,3] => 0
[5,2,4,3,1] => 0
[5,3,1,2,4] => 1
[5,3,1,4,2] => 0
[5,3,2,1,4] => 0
[5,3,2,4,1] => 0
[5,3,4,1,2] => 0
[5,3,4,2,1] => 0
[5,4,1,2,3] => 1
[5,4,1,3,2] => 0
[5,4,2,1,3] => 0
[5,4,2,3,1] => 0
[5,4,3,1,2] => 0
[5,4,3,2,1] => 0

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Created: Jul 03, 2024 at 11:46 by Elena Hoster

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Last Updated: Jul 04, 2024 at 09:15 by Christian Stump