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Identifier

St000766: Integer compositions ⟶ ℤ

Values

[1] => 0

[1,1] => 0

[2] => 0

[1,1,1] => 0

[1,2] => 0

[2,1] => 1

[3] => 0

[1,1,1,1] => 0

[1,1,2] => 0

[1,2,1] => 1

[1,3] => 0

[2,1,1] => 2

[2,2] => 0

[3,1] => 1

[4] => 0

[1,1,1,1,1] => 0

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

[1,1,2,1] => 1

[1,1,3] => 0

[1,2,1,1] => 2

[1,2,2] => 0

[1,3,1] => 1

[1,4] => 0

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

[2,1,2] => 1

[2,2,1] => 2

[2,3] => 0

[3,1,1] => 2

[3,2] => 1

[4,1] => 1

[5] => 0

[1,1,1,1,1,1] => 0

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

[1,1,1,2,1] => 1

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

[1,1,2,1,1] => 2

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

[1,1,3,1] => 1

[1,1,4] => 0

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

[1,2,1,2] => 1

[1,2,2,1] => 2

[1,2,3] => 0

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

[1,3,2] => 1

[1,4,1] => 1

[1,5] => 0

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

[2,1,1,2] => 2

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

[2,1,3] => 1

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

[2,2,2] => 0

[2,3,1] => 2

[2,4] => 0

[3,1,1,1] => 3

[3,1,2] => 2

[3,2,1] => 3

[3,3] => 0

[4,1,1] => 2

[4,2] => 1

[5,1] => 1

[6] => 0

[1,1,1,1,1,1,1] => 0

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

[1,1,1,1,2,1] => 1

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

[1,1,1,2,1,1] => 2

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

[1,1,1,3,1] => 1

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

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

[1,1,2,1,2] => 1

[1,1,2,2,1] => 2

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

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

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

[1,1,4,1] => 1

[1,1,5] => 0

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

[1,2,1,1,2] => 2

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

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

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

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

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

[1,2,4] => 0

[1,3,1,1,1] => 3

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

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

[1,3,3] => 0

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

[1,4,2] => 1

[1,5,1] => 1

[1,6] => 0

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

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

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

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

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

[2,1,2,2] => 1

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Description

The number of inversions of an integer composition.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.

References

[1] MacMahon, P. A. Two Applications of General Theorems in Combinatory Analysis: (1) To the Theory of Inversions of Permutations; (2) To the Ascertainment of the Numbers of Terms in the Development of a Determinant which has Amongst its Elements an Arbitrary Number of Zeros MathSciNet:1576566

Code

def statistic(w):
    return len([(i,j) for i in range(len(w)) for j in range(i, len(w)) if w[i] > w[j]])

Created

Apr 09, 2017 at 14:37 by Martin Rubey

Updated

Jan 11, 2018 at 09:38 by Martin Rubey