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Identifier

St000030: Permutations ⟶ ℤ

Values

[1] => 0

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,4,1,5,2] => 6

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

[3,4,2,5,1] => 6

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

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

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

[3,5,1,4,2] => 6

>>> Load all 873 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 3\ q^{2}$

$F_{4} = 1 + 3\ q + 7\ q^{2} + 9\ q^{3} + 4\ q^{4}$

$F_{5} = 1 + 4\ q + 12\ q^{2} + 24\ q^{3} + 35\ q^{4} + 24\ q^{5} + 20\ q^{6}$

$F_{6} = 1 + 5\ q + 18\ q^{2} + 46\ q^{3} + 93\ q^{4} + 137\ q^{5} + 148\ q^{6} + 136\ q^{7} + 100\ q^{8} + 36\ q^{9}$

Description

The sum of the descent differences of a permutations.
This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.

References

[1] Babson, E., Steingr'ımsson, E. Generalized permutation patterns and a classification of the Mahonian statistics MathSciNet:1758852
[2] Petersen, T. K., Tenner, B. E. The depth of a permutation arXiv:1202.4765
[3] Triangle read by rows: For n >= 1, k >= 0, T(n,k) = the number of permutations pi of n such that the total distance sum_i abs(i-pi(i)) = 2k. Equivalently, k = sum_i max(i-pi(i),0). OEIS:A062869

Code

def statistic(x):
    return sum(x[i]-x[i+1] for i in range(x.size()-1) if x[i]>x[i+1])

Created

Nov 30, 2012 at 21:30 by Chris Berg

Updated

May 29, 2015 at 16:16 by Martin Rubey