Identifier
Values
[1,0] => [(1,2)] => [2,1] => [1,2] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [1,4,2,3] => 1
[1,1,0,0] => [(1,4),(2,3)] => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,4,2,3,6,5] => 1
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,6,2,3,4,5] => 1
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,6,2,3,4,5] => 1
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,2,5,3,4,6] => 1
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 1
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [1,8,2,3,4,5,6,7] => 1
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [1,8,2,3,4,5,6,7] => 1
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 1
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 1
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 1
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 1
[1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => 1
[1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => [4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => 1
[1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => [6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => 1
[1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => [8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 1
[1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 1
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 first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect 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.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.