Identifier
Values
[1] => [1,0] => [(1,2)] => [2,1] => 1
[1,1] => [1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 2
[2] => [1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => 1
[1,1,1] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 3
[1,2] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 2
[2,1] => [1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => 1
[3] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => 4
[2,2] => [1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
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Description
The size of the overlap set of a permutation.
For a permutation $\pi\in\mathfrak S_n$ this is the number of indices $i < n$ such that the standardisation of $\pi_1\dots\pi_{n-i}$ equals the standardisation of $\pi_{i+1}\dots\pi_n$. In particular, for $n > 1$, the statistic is at least one, because the standardisations of $\pi_1$ and $\pi_n$ are both $1$.
For example, for $\pi=2143$, the standardisations of $21$ and $43$ are equal, and so are the standardisations of $2$ and $3$. Thus, the statistic on $\pi$ is $2$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given 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.