Identifier
-
Mp00159:
Permutations
—Demazure product with inverse⟶
Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤ (values match St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.)
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 2
[2,1,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 2
[2,1,3,4] => [2,1,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
[2,3,1,4] => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 2
[2,4,1,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 2
[3,1,2,4] => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 2
[3,2,1,4] => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,3,5,2,4] => [1,4,5,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 3
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => 2
[2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 3
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 2
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 2
[2,3,5,1,4] => [4,2,5,1,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,4,1,3,5] => [3,4,1,2,5] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => 3
[2,4,3,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,4,5,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,5,3,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,5,4,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,1,2,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 3
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 2
[3,1,4,2,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 2
[3,1,5,2,4] => [4,2,5,1,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,2,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 3
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 2
[3,2,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 2
[3,2,5,1,4] => [4,2,5,1,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,4,1,2,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,4,2,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,5,2,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[4,1,2,3,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 2
[4,1,3,2,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 2
[4,2,1,3,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[4,2,3,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[4,3,1,2,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0] => 3
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,4,5,3,2,6] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,3,4,6,2,5] => [1,5,3,6,2,4] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,3,5,6,2,4] => [1,5,6,4,2,3] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,4,5,3,2,6] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,4,5,3,2,6] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [5,6,1,3,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,4,6,2,3,5] => [1,5,6,4,2,3] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,4,6,3,2,5] => [1,5,6,4,2,3] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2
[2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0] => 2
[2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
[2,1,4,5,3,6] => [2,1,5,4,3,6] => [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[2,1,4,6,3,5] => [2,1,5,6,3,4] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[2,1,5,3,4,6] => [2,1,5,4,3,6] => [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[2,1,5,4,3,6] => [2,1,5,4,3,6] => [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[2,3,1,4,5,6] => [3,2,1,4,5,6] => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0] => 4
[2,3,1,4,6,5] => [3,2,1,4,6,5] => [3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0] => 3
[2,3,1,5,4,6] => [3,2,1,5,4,6] => [3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0] => 3
[2,3,1,5,6,4] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
[2,3,1,6,4,5] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
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Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
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
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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