- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 6\ q + 6\ q^{2} + q^{3}$

$F_{5} = 1 + 10\ q + 21\ q^{2} + 9\ q^{3} + q^{4}$

$F_{6} = 1 + 15\ q + 53\ q^{2} + 48\ q^{3} + 14\ q^{4} + q^{5}$

Description

The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map

$\Phi$ in [1, sec.3].

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.).