Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000789: Perfect matchings ⟶ ℤ
Values
[1] => [1,0] => [1,0] => [(1,2)] => 1
[2] => [1,0,1,0] => [1,1,0,0] => [(1,4),(2,3)] => 1
[1,1] => [1,1,0,0] => [1,0,1,0] => [(1,2),(3,4)] => 1
[3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 1
[2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 2
[1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 3
[2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 4
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 3
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [(1,10),(2,3),(4,5),(6,9),(7,8)] => 3
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [(1,10),(2,3),(4,9),(5,8),(6,7)] => 2
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [(1,6),(2,3),(4,5),(7,8),(9,10)] => 6
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 6
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [(1,10),(2,3),(4,9),(5,6),(7,8)] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => 4
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => 1
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => 3
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => 1
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => 4
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => 1
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Description
The number of crossing-similar perfect matchings of a perfect matching.
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of crossings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are crossing-similar, it is enough to check that $M$ and $N$ have the same number of crossings, and that the distribution of crossings agrees for their direct children.
[thm 3.3, 1], see also [2], gives the number of equivalence classes of crossing-similar matchings with $n$ arcs as $$2^{n-2}\left(\binom{n}{2}+2\right).$$
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of crossings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are crossing-similar, it is enough to check that $M$ and $N$ have the same number of crossings, and that the distribution of crossings agrees for their direct children.
[thm 3.3, 1], see also [2], gives the number of equivalence classes of crossing-similar matchings with $n$ arcs as $$2^{n-2}\left(\binom{n}{2}+2\right).$$
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
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.
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