Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000029: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1] => [1] => 0
{{1,2}} => [2,1] => [1,2] => [1,2] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}} => [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}} => [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}} => [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}} => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}} => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
{{1,3},{2,4}} => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,3,4}} => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,3,4,5,2] => [1,5,3,4,2] => 3
{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,3,5,2,4] => [1,4,3,5,2] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,3,5,2,4] => [1,4,3,5,2] => 3
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,4,5,2,3] => [1,3,5,4,2] => 3
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 3
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,4,5,2,3] => [1,3,5,4,2] => 3
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,4,2,5,3] => [1,4,5,2,3] => 4
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,3,2] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,5,2,3,4] => [1,3,4,5,2] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 3
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 2
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 2
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 3
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 2
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 2
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => 3
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => 3
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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].
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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
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