- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000325: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000470The number of runs in a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 3

[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] => 3

[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] => 3

[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] => 2

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

[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] => 3

[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] => 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] => 2

[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] => 2

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 4

[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] => 4

[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] => 4

[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] => 3

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

[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] => 4

[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] => 3

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

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

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

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

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

[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] => 2

[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] => 2

[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] => 4

[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] => 3

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

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$F_{2} = 2\ q$

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

$F_{4} = 2\ q + 8\ q^{2} + 4\ q^{3}$

$F_{5} = 2\ q + 14\ q^{2} + 21\ q^{3} + 5\ q^{4}$

$F_{6} = 2\ q + 22\ q^{2} + 56\ q^{3} + 46\ q^{4} + 6\ q^{5}$

$F_{7} = 2\ q + 32\ q^{2} + 123\ q^{3} + 173\ q^{4} + 92\ q^{5} + 7\ q^{6}$

Description

The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents St000021The number of descents of a permutation.. This also matches the number of runs in a permutation St000470The number of runs in a permutation..
See also St000308The height of the tree associated to a permutation. for the height of this tree.

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.