- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ 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] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

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

$F_{4} = 2 + 8\ q + 4\ q^{2}$

$F_{5} = 2 + 14\ q + 21\ q^{2} + 5\ q^{3}$

$F_{6} = 2 + 22\ q + 56\ q^{2} + 46\ q^{3} + 6\ q^{4}$

$F_{7} = 2 + 32\ q + 123\ q^{2} + 173\ q^{3} + 92\ q^{4} + 7\ q^{5}$

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

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 non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

restriction

Description

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