Identifier
-
Mp00146:
Dyck paths
—to tunnel matching⟶
Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001444: Perfect matchings ⟶ ℤ
Values
[1,0] => [(1,2)] => [(1,2)] => 0
[1,0,1,0] => [(1,2),(3,4)] => [(1,2),(3,4)] => 0
[1,1,0,0] => [(1,4),(2,3)] => [(1,3),(2,4)] => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [(1,2),(3,4),(5,6)] => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [(1,2),(3,5),(4,6)] => 1
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [(1,3),(2,4),(5,6)] => 1
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [(1,3),(2,5),(4,6)] => 1
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [(1,4),(2,5),(3,6)] => 1
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [(1,2),(3,4),(5,6),(7,8)] => 0
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [(1,2),(3,4),(5,7),(6,8)] => 1
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [(1,2),(3,5),(4,6),(7,8)] => 1
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [(1,2),(3,5),(4,7),(6,8)] => 1
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [(1,2),(3,6),(4,7),(5,8)] => 1
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => [(1,3),(2,4),(5,6),(7,8)] => 1
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [(1,3),(2,4),(5,7),(6,8)] => 2
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => [(1,3),(2,5),(4,6),(7,8)] => 1
[1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => [(1,3),(2,5),(4,7),(6,8)] => 2
[1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => [(1,3),(2,6),(4,7),(5,8)] => 2
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [(1,4),(2,5),(3,6),(7,8)] => 1
[1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => [(1,4),(2,5),(3,7),(6,8)] => 2
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => [(1,4),(2,6),(3,7),(5,8)] => 1
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [(1,5),(2,6),(3,7),(4,8)] => 2
[1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => 0
[1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => [(1,2),(3,4),(5,6),(7,9),(8,10)] => 1
[1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10)] => [(1,2),(3,4),(5,7),(6,8),(9,10)] => 1
[1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => [(1,2),(3,4),(5,7),(6,9),(8,10)] => 1
[1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => [(1,2),(3,4),(5,8),(6,9),(7,10)] => 1
[1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => [(1,2),(3,5),(4,6),(7,8),(9,10)] => 1
[1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => [(1,2),(3,5),(4,6),(7,9),(8,10)] => 2
[1,0,1,1,0,1,0,0,1,0] => [(1,2),(3,8),(4,5),(6,7),(9,10)] => [(1,2),(3,5),(4,7),(6,8),(9,10)] => 1
[1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => [(1,2),(3,5),(4,7),(6,9),(8,10)] => 2
[1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => [(1,2),(3,5),(4,8),(6,9),(7,10)] => 2
[1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => [(1,2),(3,6),(4,7),(5,8),(9,10)] => 1
[1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => [(1,2),(3,6),(4,7),(5,9),(8,10)] => 2
[1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => [(1,2),(3,6),(4,8),(5,9),(7,10)] => 1
[1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => [(1,2),(3,7),(4,8),(5,9),(6,10)] => 2
[1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => [(1,3),(2,4),(5,6),(7,8),(9,10)] => 1
[1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => [(1,3),(2,4),(5,6),(7,9),(8,10)] => 2
[1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => [(1,3),(2,4),(5,7),(6,8),(9,10)] => 2
[1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => [(1,3),(2,4),(5,7),(6,9),(8,10)] => 2
[1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => [(1,3),(2,4),(5,8),(6,9),(7,10)] => 2
[1,1,0,1,0,0,1,0,1,0] => [(1,6),(2,3),(4,5),(7,8),(9,10)] => [(1,3),(2,5),(4,6),(7,8),(9,10)] => 1
[1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => [(1,3),(2,5),(4,6),(7,9),(8,10)] => 2
[1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => [(1,3),(2,5),(4,7),(6,8),(9,10)] => 2
[1,1,0,1,0,1,0,1,0,0] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => [(1,3),(2,5),(4,7),(6,9),(8,10)] => 2
[1,1,0,1,0,1,1,0,0,0] => [(1,10),(2,3),(4,5),(6,9),(7,8)] => [(1,3),(2,5),(4,8),(6,9),(7,10)] => 2
[1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => [(1,3),(2,6),(4,7),(5,8),(9,10)] => 2
[1,1,0,1,1,0,0,1,0,0] => [(1,10),(2,3),(4,7),(5,6),(8,9)] => [(1,3),(2,6),(4,7),(5,9),(8,10)] => 2
[1,1,0,1,1,0,1,0,0,0] => [(1,10),(2,3),(4,9),(5,6),(7,8)] => [(1,3),(2,6),(4,8),(5,9),(7,10)] => 2
[1,1,0,1,1,1,0,0,0,0] => [(1,10),(2,3),(4,9),(5,8),(6,7)] => [(1,3),(2,7),(4,8),(5,9),(6,10)] => 2
[1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => [(1,4),(2,5),(3,6),(7,8),(9,10)] => 1
[1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => [(1,4),(2,5),(3,6),(7,9),(8,10)] => 2
[1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => [(1,4),(2,5),(3,7),(6,8),(9,10)] => 2
[1,1,1,0,0,1,0,1,0,0] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => [(1,4),(2,5),(3,7),(6,9),(8,10)] => 2
[1,1,1,0,0,1,1,0,0,0] => [(1,10),(2,5),(3,4),(6,9),(7,8)] => [(1,4),(2,5),(3,8),(6,9),(7,10)] => 2
[1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => [(1,4),(2,6),(3,7),(5,8),(9,10)] => 1
[1,1,1,0,1,0,0,1,0,0] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => [(1,4),(2,6),(3,7),(5,9),(8,10)] => 2
[1,1,1,0,1,0,1,0,0,0] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => [(1,4),(2,6),(3,8),(5,9),(7,10)] => 2
[1,1,1,0,1,1,0,0,0,0] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => [(1,4),(2,7),(3,8),(5,9),(6,10)] => 2
[1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => [(1,5),(2,6),(3,7),(4,8),(9,10)] => 2
[1,1,1,1,0,0,0,1,0,0] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => [(1,5),(2,6),(3,7),(4,9),(8,10)] => 2
[1,1,1,1,0,0,1,0,0,0] => [(1,10),(2,9),(3,6),(4,5),(7,8)] => [(1,5),(2,6),(3,8),(4,9),(7,10)] => 2
[1,1,1,1,0,1,0,0,0,0] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => [(1,5),(2,7),(3,8),(4,9),(6,10)] => 2
[1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [(1,6),(2,7),(3,8),(4,9),(5,10)] => 2
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Description
The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching.
Two pairs $(a,b)$ and $(c,d)$ (with $a < b$ and $c < d$) in a perfect matching cross if and only if $a < c < b < d$ or $c < a < d < b$. Define a skew symmetric matrix $M$ whose rows and columns are indexed by the pairs of the matching, with
$$ M_{(a,b),(c,d)} = \begin{cases} 1 &\text{if \(a < c < b < d\)}\\ -1 &\text{if \(c < a < d < b\)}\\ 0 &\text{otherwise} \end{cases} $$
The rank of this matrix is always even. The present statistic is half of the matrix' rank.
Two pairs $(a,b)$ and $(c,d)$ (with $a < b$ and $c < d$) in a perfect matching cross if and only if $a < c < b < d$ or $c < a < d < b$. Define a skew symmetric matrix $M$ whose rows and columns are indexed by the pairs of the matching, with
$$ M_{(a,b),(c,d)} = \begin{cases} 1 &\text{if \(a < c < b < d\)}\\ -1 &\text{if \(c < a < d < b\)}\\ 0 &\text{otherwise} \end{cases} $$
The rank of this matrix is always even. The present statistic is half of the matrix' rank.
Map
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.
This yields the perfect matching with the number of nestings and crossings exchanged.
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.
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