Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001171: Permutations ⟶ ℤ (values match St000055The inversion sum of a permutation.)
Values
[1,0] => [] => [] => [1] => 0
[1,0,1,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,0,0] => [] => [] => [1] => 0
[1,0,1,0,1,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,0,1,1,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,0,0,1,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,0,1,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,0,0,0] => [] => [] => [1] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,0,1,1,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,0,0,1,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,0,1,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,1,0,0,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => [] => [1] => 0
[] => [] => [] => [1] => 0
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 5
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => [3,1,2] => 3
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0] => [] => [] => [1] => 0
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 6
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
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Description
The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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