Identifier
-
Mp00307:
Posets
—promotion cycle type⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000689: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0] => 0
([],2) => [2] => [1,0,1,0] => 1
([(0,1)],2) => [1] => [1,0] => 0
([],3) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(1,2)],3) => [3] => [1,0,1,0,1,0] => 2
([(0,1),(0,2)],3) => [2] => [1,0,1,0] => 1
([(0,2),(2,1)],3) => [1] => [1,0] => 0
([(0,2),(1,2)],3) => [2] => [1,0,1,0] => 1
([(0,1),(0,2),(0,3)],4) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,2),(0,3),(3,1)],4) => [3] => [1,0,1,0,1,0] => 2
([(0,1),(0,2),(1,3),(2,3)],4) => [2] => [1,0,1,0] => 1
([(1,2),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,3),(3,1),(3,2)],4) => [2] => [1,0,1,0] => 1
([(0,3),(1,3),(3,2)],4) => [2] => [1,0,1,0] => 1
([(0,3),(1,3),(2,3)],4) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,3),(1,2)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,3),(1,2),(1,3)],4) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,1,0,0,0] => 0
([(0,3),(2,1),(3,2)],4) => [1] => [1,0] => 0
([(0,3),(1,2),(2,3)],4) => [3] => [1,0,1,0,1,0] => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [2] => [1,0,1,0] => 1
([(0,3),(0,4),(3,2),(4,1)],5) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [2,2] => [1,1,1,0,0,0] => 0
([(0,4),(4,1),(4,2),(4,3)],5) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,4),(1,4),(4,2),(4,3)],5) => [2,2] => [1,1,1,0,0,0] => 0
([(0,4),(1,4),(2,4),(4,3)],5) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,4),(1,4),(2,3),(4,2)],5) => [2] => [1,0,1,0] => 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [2,2] => [1,1,1,0,0,0] => 0
([(0,2),(0,4),(3,1),(4,3)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [3] => [1,0,1,0,1,0] => 2
([(1,4),(3,2),(4,3)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 4
([(0,3),(3,4),(4,1),(4,2)],5) => [2] => [1,0,1,0] => 1
([(0,4),(1,2),(2,4),(4,3)],5) => [3] => [1,0,1,0,1,0] => 2
([(0,4),(3,2),(4,1),(4,3)],5) => [3] => [1,0,1,0,1,0] => 2
([(0,4),(2,3),(3,1),(4,2)],5) => [1] => [1,0] => 0
([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,4),(1,2),(2,3),(3,4)],5) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [2] => [1,0,1,0] => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [2,2] => [1,1,1,0,0,0] => 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [2,2] => [1,1,1,0,0,0] => 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [2,2] => [1,1,1,0,0,0] => 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [2] => [1,0,1,0] => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [2,2] => [1,1,1,0,0,0] => 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6) => [2,2] => [1,1,1,0,0,0] => 0
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [3] => [1,0,1,0,1,0] => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [2] => [1,0,1,0] => 1
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 0
([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 4
([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [2] => [1,0,1,0] => 1
([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [2] => [1,0,1,0] => 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6) => [2,2] => [1,1,1,0,0,0] => 0
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [3] => [1,0,1,0,1,0] => 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => [1,0] => 0
([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [3] => [1,0,1,0,1,0] => 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => 4
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [2] => [1,0,1,0] => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [3] => [1,0,1,0,1,0] => 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7) => [2,2] => [1,1,1,0,0,0] => 0
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7) => [2,2] => [1,1,1,0,0,0] => 0
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [2,2] => [1,1,1,0,0,0] => 0
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7) => [2,2] => [1,1,1,0,0,0] => 0
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [5] => [1,0,1,0,1,0,1,0,1,0] => 4
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [2] => [1,0,1,0] => 1
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [2] => [1,0,1,0] => 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7) => [2,2] => [1,1,1,0,0,0] => 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7) => [2,2] => [1,1,1,0,0,0] => 0
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => [3,3] => [1,1,1,0,1,0,0,0] => 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [3,2] => [1,0,1,1,1,0,0,0] => 0
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => [4] => [1,0,1,0,1,0,1,0] => 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [2] => [1,0,1,0] => 1
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 0
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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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
promotion cycle type
Description
The cycle type of promotion on the linear extensions of a poset.
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