Identifier
Values
[1,0] => [(1,2)] => [2,1] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 2
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => 2
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => 2
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => 2
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => 3
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The maximal number of nonzero entries on a diagonal of a permutation matrix.
For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right.
In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$
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
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.