Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
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
[1,1,0,0] => [1,2] => [1,2] => [1,0,1,0] => 2
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,5,1,2,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,2,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [3,4,1,2,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,3,4,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [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
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searching the database for the individual values of this statistic
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
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 $\pi = \pi^{-1}$.
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
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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