Identifier
-
Mp00146:
Dyck paths
—to tunnel matching⟶
Perfect matchings
St000044: Perfect matchings ⟶ ℤ
Values
[1,0] => [(1,2)] => 2
[1,0,1,0] => [(1,2),(3,4)] => 3
[1,1,0,0] => [(1,4),(2,3)] => 3
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => 4
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => 4
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => 4
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => 4
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => 5
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => 5
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => 5
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => 5
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => 5
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => 5
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => 5
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => 5
[1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => 5
[1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => 5
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => 5
[1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => 5
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => 5
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => 5
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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
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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