- 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
St000325: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000470The number of runs in a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

[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] => 5

[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] => 4

[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] => 4

[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] => 4

[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] => 3

[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] => 4

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 4

[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] => 3

[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] => 2

[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] => 4

[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] => 3

[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] => 3

[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] => 4

[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] => 2

[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] => 3

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 2

[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] => 3

[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] => 4

[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] => 2

[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] => 4

[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] => 2

[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] => 2

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 1

[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] => 6

[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] => 5

[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] => 5

[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] => 5

[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] => 4

[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] => 5

[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] => 4

[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] => 4

[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] => 4

[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] => 4

[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] => 4

[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] => 5

[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] => 4

[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] => 3

[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] => 5

[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] => 4

[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] => 4

[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] => 4

[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] => 3

[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] => 4

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 3

[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] => 4

[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] => 4

[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] => 3

[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] => 4

[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] => 3

[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] => 3

[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] => 3

[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] => 4

[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] => 4

[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] => 4

[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] => 4

>>> Load all 196 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

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

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

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

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

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.).