Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤ
Values
[1,0,1,0] => [1] => [1,0] => [(1,2)] => 1
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[1,0,1,1,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 1
[1,1,0,0,1,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 1
[1,1,0,1,0,0] => [1] => [1,0] => [(1,2)] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 1
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => 1
[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 1
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 1
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[1,1,0,1,0,0,1,0] => [3,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 1
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[1,1,0,1,1,0,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 1
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 2
[1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 1
[1,1,1,0,1,0,0,0] => [1] => [1,0] => [(1,2)] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => 1
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => 1
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => 1
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)] => 1
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)] => 1
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 2
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 2
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0] => [(1,2)] => 1
[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] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => 1
[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] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => 1
[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] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => 1
[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] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)] => 1
[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] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => 1
[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] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => 1
[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] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => 1
[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] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 1
[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] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)] => 1
[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] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)] => 1
[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] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 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] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)] => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => 1
[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] => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[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] => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0] => [(1,2)] => 1
[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] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => 1
[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] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => 1
[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] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => 1
[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] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)] => 1
[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] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 1
[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] => [(1,8),(2,3),(4,5),(6,7)] => 1
[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] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => 1
[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] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => 1
[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] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 1
[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] => [(1,8),(2,7),(3,6),(4,5)] => 1
[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] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)] => 1
[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] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 1
>>> Load all 243 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
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
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
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 $(n-1,\ldots,2,1)$.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!