- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000710: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2$

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

$F_{4} = 5 + 7\ q + 2\ q^{2}$

$F_{5} = 8 + 21\ q + 13\ q^{2}$

$F_{6} = 13 + 53\ q + 61\ q^{2} + 5\ q^{3}$

Description

The number of big deficiencies of a permutation.
A big deficiency of a permutation

$\pi$ is an index

$i$ such that

$i - \pi(i) > 1$ .
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.

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

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.