Identifier
-
Mp00068:
Permutations
—Simion-Schmidt map⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ 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] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0] => 2
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
[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] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 2
[2,4,1,3] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,5,2,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[2,1,3,4,5] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,1,4,3,5] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,3,1,4,5] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[2,3,1,5,4] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[2,3,4,1,5] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,3,5,1,4] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,4,1,3,5] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[2,4,1,5,3] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[2,4,3,1,5] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,4,5,1,3] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,5,1,3,4] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[2,5,1,4,3] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[2,5,3,1,4] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[2,5,4,1,3] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,2,1,4,5] => [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 2
[3,2,4,1,5] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[3,2,5,1,4] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[3,4,2,1,5] => [3,5,2,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,5,2,1,4] => [3,5,2,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,4,3,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,4,3,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,4,5,3,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,4,5,6,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,4,6,3,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,4,6,5,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,5,3,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,5,3,6,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,5,4,3,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,5,4,6,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,5,6,3,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,5,6,4,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,6,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,6,3,5,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,6,4,3,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,6,4,5,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,6,5,3,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,6,5,4,3] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,2,4,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,2,4,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,2,5,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,2,5,6,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,2,6,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,2,6,5,4] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,4,2,5,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,4,2,6,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,4,5,2,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,4,5,6,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,4,6,2,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,4,6,5,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,3,5,2,4,6] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 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
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
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
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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