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Identifier

St000638: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 2

[1,2,3] => 1

[1,3,2] => 2

[2,1,3] => 3

[2,3,1] => 2

[3,1,2] => 3

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of up-down runs of a permutation.
An up-down run of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$.
For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and
$643$.

References

[1] Zhuang, Y. Counting permutations by runs MathSciNet:3499494
[2] Zhuang, Y. Eulerian polynomials and descent statistics arXiv:1610.07218

Code

def statistic(x):
   if len(x)>1 and x[0]>x[1]:
        return x.number_of_peaks() + x.complement().number_of_peaks() + 2
   elif len(x)>0:
        return x.number_of_peaks() + x.complement().number_of_peaks() + 1
   else:
        return 0

Created

Oct 27, 2016 at 21:11 by Yan Zhuang

Updated

Oct 27, 2016 at 21:11 by Yan Zhuang