Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤ
Values
[[1]] => [1] => [1,0,1,0] => [(1,2),(3,4)] => 0
[[1],[1]] => [1,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 0
[[2]] => [2] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 0
[[1,1]] => [2] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 0
[[1],[1],[1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 0
[[2],[1]] => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 0
[[1,1],[1]] => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 0
[[3]] => [3] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 0
[[2,1]] => [3] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 0
[[1,1,1]] => [3] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 0
[[1],[1],[1],[1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => 0
[[2],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 0
[[2],[2]] => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 0
[[1,1],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 0
[[1,1],[1,1]] => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 0
[[3],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 0
[[2,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 0
[[1,1,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 0
[[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 0
[[3,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 0
[[2,2]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 0
[[2,1,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 0
[[1,1,1,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => 0
[[2],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => 0
[[2],[2],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 0
[[1,1],[1],[1],[1]] => [2,1,1,1] => [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],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 0
[[3],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 0
[[3],[2]] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 0
[[2,1],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 0
[[2,1],[2]] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 0
[[2,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 0
[[1,1,1],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 0
[[1,1,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 0
[[4],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => 0
[[3,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => 0
[[2,2],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => 0
[[2,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => 0
[[1,1,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => 0
[[2],[2],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => 0
[[2],[2],[2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 0
[[1,1],[1,1],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => 0
[[1,1],[1,1],[1,1]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => 0
[[3],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 0
[[3],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 0
[[3],[3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 0
[[2,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 0
[[2,1],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 0
[[2,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 0
[[2,1],[2,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 0
[[1,1,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => 0
[[1,1,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 0
[[1,1,1],[1,1,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => 0
[[4],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => 0
[[4],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[3,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => 0
[[3,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[3,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[2,2],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => 0
[[2,2],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[2,2],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[2,1,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => 0
[[2,1,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[2,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[1,1,1,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => 0
[[1,1,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => 0
[[2],[2],[2],[1]] => [2,2,2,1] => [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],[1,1],[1]] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => 0
[[3],[2],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 0
[[3],[2],[2]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => 0
[[3],[3],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => 0
[[2,1],[2],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 0
[[2,1],[2],[2]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => 0
[[2,1],[1,1],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 0
[[2,1],[1,1],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => 0
[[2,1],[2,1],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => 0
[[1,1,1],[1,1],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => 0
[[1,1,1],[1,1],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => 0
[[1,1,1],[1,1,1],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => 0
[[4],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 0
[[4],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[4],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[3,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 0
[[3,1],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[3,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[3,1],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[3,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[2,2],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 0
[[2,2],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[2,2],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[2,2],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[2,1,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 0
[[2,1,1],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[2,1,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[2,1,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[2,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[1,1,1,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => 0
[[1,1,1,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => 0
[[1,1,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => 0
[[3],[2],[2],[1]] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => 0
[[3],[3],[1],[1]] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => 0
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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
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 underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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