- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000209: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 4\ q + 6\ q^{2} + 13\ q^{3}$

$F_{5} = 1 + 7\ q + 15\ q^{2} + 26\ q^{3} + 71\ q^{4}$

$F_{6} = 1 + 12\ q + 39\ q^{2} + 65\ q^{3} + 142\ q^{4} + 461\ q^{5}$

Description

Maximum difference of elements in cycles.
Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ .
The statistic is then the maximum of this value over all cycles in the permutation.

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

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.