Identifier
Values
[1,0] => [(1,2)] => [2,1] => [2,1] => 0
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => [4,1,3,2] => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,3,5,4] => 1
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,1,3,2,6,5] => 1
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,1,3,2,5,4] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,2,6,1,4,3] => 3
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 0
[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] => [2,1,4,3,6,5,8,7,10,9] => 0
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Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.
There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
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.