Identifier
-
Mp00060:
Permutations
—Robinson-Schensted tableau shape⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000044: Perfect matchings ⟶ ℤ
Values
[1] => [1] => [1,0,1,0] => [(1,2),(3,4)] => 3
[1,2] => [2] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 4
[2,1] => [1,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 4
[1,2,3] => [3] => [1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 5
[1,3,2] => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 4
[2,1,3] => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 4
[2,3,1] => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 4
[3,1,2] => [2,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 4
[3,2,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 5
[1,2,4,3] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[1,3,2,4] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[1,3,4,2] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[1,4,2,3] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[1,4,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[2,1,3,4] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 5
[2,3,1,4] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[2,3,4,1] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 5
[2,4,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[3,1,2,4] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 5
[3,2,1,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[3,2,4,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 5
[3,4,2,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[4,1,2,3] => [3,1] => [1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[4,1,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[4,2,1,3] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[4,2,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[4,3,1,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[1,2,5,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,3,2,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[1,3,5,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[1,3,5,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,4,2,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[1,4,3,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,4,3,5,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,4,5,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[1,4,5,3,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,5,2,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,5,3,2,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,5,3,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,5,4,2,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,1,3,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,1,4,3,5] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,1,4,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,1,5,3,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,1,5,4,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[2,3,1,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,3,5,1,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,3,5,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,4,1,3,5] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,4,1,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,4,3,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,4,3,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,4,5,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,4,5,3,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,5,1,3,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[2,5,1,4,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[2,5,3,1,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,5,3,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[2,5,4,1,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,1,2,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,1,4,2,5] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,1,4,5,2] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,1,5,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,1,5,4,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,2,1,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[3,2,1,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,2,4,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[3,2,4,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[3,2,5,1,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,2,5,4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,4,1,2,5] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,4,1,5,2] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,4,2,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[3,4,2,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[3,4,5,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,4,5,2,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[3,5,1,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[3,5,1,4,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,5,2,1,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,5,2,4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[3,5,4,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,1,2,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[4,1,3,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[4,1,3,5,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[4,1,5,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[4,1,5,3,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,2,1,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[4,2,1,5,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,2,3,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[4,2,3,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[4,2,5,1,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,2,5,3,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,3,1,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[4,3,1,5,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,3,5,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[4,5,1,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[4,5,1,3,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
>>> Load all 373 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of vertices of the unicellular map given by a perfect matching.
If the perfect matching of $2n$ elements is viewed as a fixed point-free involution $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$.
Example The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle.
Let $\epsilon_v(n)$ is the number of matchings of $2n$ such that yield $v$ cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers $\epsilon_v(n)$.
$$ \sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k. $$
If the perfect matching of $2n$ elements is viewed as a fixed point-free involution $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$.
Example The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle.
Let $\epsilon_v(n)$ is the number of matchings of $2n$ such that yield $v$ cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers $\epsilon_v(n)$.
$$ \sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k. $$
Map
Robinson-Schensted tableau shape
Description
Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.
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 Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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!