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Identifier

St000619: Permutations ⟶ ℤ

Values

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

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$F_{2} = 2\ q$

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

$F_{4} = 4\ q + 16\ q^{2} + 4\ q^{3}$

$F_{5} = 5\ q + 55\ q^{2} + 55\ q^{3} + 5\ q^{4}$

$F_{6} = 6\ q + 156\ q^{2} + 396\ q^{3} + 156\ q^{4} + 6\ q^{5}$

Description

The number of cyclic descents of a permutation.
For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is given by the number of indices

$1 \leq i \leq n$ such that

$\pi(i) > \pi(i+1)$ where we set

$\pi(n+1) = \pi(1)$ .

References

[1] Elizalde, S., Roichman, Y. On rotated Schur-positive sets arXiv:1609.07335

Code

def statistic(pi):
    if pi[-1] > pi[0]:
        cyc = 1
    else:
        cyc = 0
    return pi.number_of_descents() + cyc

Created

Oct 11, 2016 at 13:51 by Christian Stump

Updated

Oct 11, 2016 at 16:01 by Christian Stump