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Identifier

St001859: Permutations ⟶ ℤ

Values

[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] => 1

[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] => 1

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 5\ q$

$F_{4} = 1 + 22\ q + q^{2}$

$F_{5} = 1 + 110\ q + 9\ q^{2}$

$F_{6} = 1 + 659\ q + 59\ q^{2} + q^{3}$

Description

The number of factors of the Stanley symmetric function associated with a permutation.
For example, the Stanley symmetric function of

$\pi=321645$ equals

$20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$

Code

def statistic(pi):
    return sum(m for _, m in pi.stanley_symmetric_function().factor())

Created

Nov 29, 2022 at 12:32 by Martin Rubey

Updated

Nov 29, 2022 at 12:32 by Martin Rubey