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Identifier

St001673: Integer compositions ⟶ ℤ

Values

[1] => 0

[1,1] => 0

[2] => 0

[1,1,1] => 0

[1,2] => 1

[2,1] => 1

[3] => 0

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

[1,1,2] => 1

[1,2,1] => 0

[1,3] => 1

[2,1,1] => 1

[2,2] => 0

[3,1] => 1

[4] => 0

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

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

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

[1,1,3] => 1

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

[1,2,2] => 1

[1,3,1] => 0

[1,4] => 1

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

[2,1,2] => 0

[2,2,1] => 1

[2,3] => 1

[3,1,1] => 1

[3,2] => 1

[4,1] => 1

[5] => 0

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

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

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

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

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

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

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

[1,1,4] => 1

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

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

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

[1,2,3] => 1

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

[1,3,2] => 1

[1,4,1] => 0

[1,5] => 1

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

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

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

[2,1,3] => 1

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

[2,2,2] => 0

[2,3,1] => 1

[2,4] => 1

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

[3,1,2] => 1

[3,2,1] => 1

[3,3] => 0

[4,1,1] => 1

[4,2] => 1

[5,1] => 1

[6] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,5] => 1

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

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

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

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

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

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

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

[1,2,4] => 1

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

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

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

[1,3,3] => 1

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

[1,4,2] => 1

[1,5,1] => 0

[1,6] => 1

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

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

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

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

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

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

>>> Load all 1023 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

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

$F_{4} = 4 + 4\ q$

$F_{5} = 4 + 12\ q$

$F_{6} = 8 + 20\ q + 4\ q^{2}$

$F_{7} = 8 + 44\ q + 12\ q^{2}$

$F_{8} = 16 + 68\ q + 44\ q^{2}$

$F_{9} = 16 + 132\ q + 100\ q^{2} + 8\ q^{3}$

$F_{10} = 32 + 196\ q + 252\ q^{2} + 32\ q^{3}$

Description

The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.

Code

def statistic(c):
    return (len(c)-sum(1 for i,j in zip(c, c[::-1]) if i == j))/2

Created

Feb 03, 2021 at 12:41 by Martin Rubey

Updated

Feb 03, 2021 at 12:41 by Martin Rubey