- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000213: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 5

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 5

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 5

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 5

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 5

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 5

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => 5

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => 5

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 5

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$F_{1} = q$

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

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

$F_{4} = 6\ q^{3} + 8\ q^{4}$

$F_{5} = 2\ q^{3} + 24\ q^{4} + 16\ q^{5}$

$F_{6} = 20\ q^{4} + 80\ q^{5} + 32\ q^{6}$

Description

The number of weak exceedances (also weak excedences) of a permutation.
This is defined as

$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$
The number of weak exceedances is given by the number of exceedances (see St000155The number of exceedances (also excedences) of a permutation.) plus the number of fixed points (see St000022The number of fixed points of a permutation.) of

$\sigma$ .

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.