Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤ
Values
[1,0] => [2,1] => [2] => [1,1,0,0,1,0] => 0
[1,0,1,0] => [3,1,2] => [3] => [1,1,1,0,0,0,1,0] => 0
[1,1,0,0] => [2,3,1] => [3] => [1,1,1,0,0,0,1,0] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
[1,1,1,0,0,0] => [2,3,4,1] => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => 0
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 0
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 0
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 0
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 0
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [6,3,7,1,2,4,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [6,3,5,1,2,7,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 0
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [8,4,1,2,7,3,5,6] => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0] => [5,9,1,2,3,8,4,6,7] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => 0
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0] => [8,4,1,2,9,3,5,6,7] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => 0
[] => [1] => [1] => [1,0,1,0] => 1
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Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path.. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
The correspondence between LNakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path.. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Map
cycle type
Description
The cycle type of a permutation as a partition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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