- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2$

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

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

$F_{5} = 2 + 7\ q + 14\ q^{2} + 19\ q^{3}$

Description

The number of Bruhat lower covers of a permutation.
This is, for a signed permutation

$\pi$ , the number of signed permutations

$\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from

$\pi$ .

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

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

Map

to signed permutation

Description

The signed permutation with all signs positive.