Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤ
Values
[1,0,1,0] => [1] => [1,0] => [(1,2)] => 0
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 0
[1,0,1,1,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 0
[1,1,0,0,1,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 0
[1,1,0,1,0,0] => [1] => [1,0] => [(1,2)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 0
[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)] => 0
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 0
[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)] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 0
[1,1,0,1,1,0,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 0
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 0
[1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 0
[1,1,1,0,1,0,0,0] => [1] => [1,0] => [(1,2)] => 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] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 0
[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)] => 0
[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)] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 0
[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)] => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0] => [(1,2)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,1,0,0] => [(1,4),(2,3)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => [(1,2),(3,4)] => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0] => [(1,2)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
[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)] => 0
>>> Load all 243 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of flips required to make a perfect matching noncrossing.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
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 $(n-1,\ldots,2,1)$.
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