- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001480: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q$

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

$F_{4} = 1 + 3\ q + 8\ q^{2} + 12\ q^{3}$

$F_{5} = 1 + 4\ q + 15\ q^{2} + 40\ q^{3} + 60\ q^{4}$

$F_{6} = 1 + 5\ q + 24\ q^{2} + 90\ q^{3} + 240\ q^{4} + 360\ q^{5}$

Description

The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.

Map

switch returns and last double rise

Description

An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.

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.