Identifier
-
Mp00090:
Permutations
—cycle-as-one-line notation⟶
Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001431: Dyck paths ⟶ ℤ
Values
[1,2] => [1,2] => [1,0,1,0] => [1,1,0,0] => 1
[2,1] => [1,2] => [1,0,1,0] => [1,1,0,0] => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[1,3,2] => [1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[2,1,3] => [1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[3,2,1] => [1,3,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
[2,4,1,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 1
[3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 1
[3,4,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[3,4,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[4,2,1,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[4,3,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,5,3,2,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,5,3,4,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,5,4,2,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,5,4,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,5,1,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,5,3,1,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,5,4,1,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,1,2,5,4] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,1,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1
[3,1,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,1,5,4,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,2,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,2,1,5,4] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,2,4,1,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1
[3,2,4,5,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[3,2,5,1,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,2,5,4,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,4,1,2,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,4,1,5,2] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,4,2,1,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,4,2,5,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,4,5,1,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,4,5,2,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[3,5,1,4,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[3,5,2,1,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
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Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
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.
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.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
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