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Identifier

St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)

Values

[] => 0

[1] => 0

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 874 entries. <<<

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Description

The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

References

[1] /Permutations/Descents
[2] Triangle of Eulerian numbers T(n,k) (n>=1, 1 <= k <= n) read by rows. OEIS:A008292
[3] Brändén, P., Claesson, A. Mesh patterns and the expansion of permutation statistics as sums of permutation patterns arXiv:1102.4226

Code

def statistic(x):
    return x.number_of_descents()

Created

Oct 11, 2011 at 19:06 by Christian Stump

Updated

May 10, 2019 at 17:11 by Henning Ulfarsson