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Identifier

St000133: Permutations ⟶ ℤ

Values

[1] => 0

[1,2] => 1

[2,1] => 0

[1,2,3] => 3

[1,3,2] => 2

[2,1,3] => 1

[2,3,1] => 0

[3,1,2] => 1

[3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => 10

[1,2,3,5,4] => 9

[1,2,4,3,5] => 8

[1,2,4,5,3] => 7

[1,2,5,3,4] => 8

[1,2,5,4,3] => 7

[1,3,2,4,5] => 7

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

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

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

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

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

[1,4,2,3,5] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 6 + 6\ q + 4\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{5} = 24 + 24\ q + 18\ q^{2} + 18\ q^{3} + 14\ q^{4} + 8\ q^{5} + 6\ q^{6} + 4\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = 120 + 120\ q + 96\ q^{2} + 96\ q^{3} + 84\ q^{4} + 66\ q^{5} + 46\ q^{6} + 30\ q^{7} + 22\ q^{8} + 16\ q^{9} + 10\ q^{10} + 6\ q^{11} + 4\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

Description

The "bounce" of a permutation.

Code

def statistic(w):
    n = len(w)
    v = w.inverse()
    bounces = [v[0]]
    while bounces[-1] < n:
        bounces.append(max(v[:bounces[-1] + 1]))
    return sum([n - i for i in bounces])

Created

Jun 19, 2013 at 15:40 by Sam Clearman

Updated

Mar 28, 2015 at 22:46 by Martin Rubey