Identifier
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Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000246: Permutations ⟶ ℤ (values match St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.)
Values
[1,0] => [] => [] => [1] => 0
[1,0,1,0] => [1] => [1,0] => [2,1] => 0
[1,1,0,0] => [] => [] => [1] => 0
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,0,1,1,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,0,0,1,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,0,1,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,0,0,0] => [] => [] => [1] => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 8
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4
[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 8
[1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 4
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 6
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,0,1,0,0,1,0] => [3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 6
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,0,1,1,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,0,1,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,0,0,0,0] => [] => [] => [1] => 0
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 8
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 7
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 7
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 6
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 10
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 6
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 5
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 8
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 8
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 4
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 6
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 2
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 10
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 6
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 10
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 6
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 6
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,0,0,0,0,0] => [] => [] => [1] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 12
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 7
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 4
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 8
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 7
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 7
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 10
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 6
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 10
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 6
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 5
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 8
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 8
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 4
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 3
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 6
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 15
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 10
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 6
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 15
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 10
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 6
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 10
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 6
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [] => [] => [1] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 10
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 4
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 12
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 7
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 15
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 4
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 8
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 7
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 7
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 10
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 6
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 10
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 6
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 8
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 4
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Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
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
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 $(n-1,\ldots,2,1)$.
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