Identifier
Values
[(1,2)] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => 2
[(1,3),(2,4)] => [3,4,1,2] => 4
[(1,4),(2,3)] => [4,3,2,1] => 4
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => 5
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => 5
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => 7
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => 7
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => 9
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => 9
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => 9
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => 7
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => 5
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => 5
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => 7
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => 9
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => 9
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => 9
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Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called almost-increasing in [5].
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.