Identifier
Values
[1,0] => [1,0] => [1,0] => [2,1] => 1
[1,0,1,0] => [1,1,0,0] => [1,0,1,0] => [3,1,2] => 2
[1,1,0,0] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 3
[1,1,0,0,1,0] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,0,0,0] => [1,1,0,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => 4
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 4
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 4
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 5
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 5
[1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 4
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 4
[1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 4
[1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 5
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 6
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 5
[1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 6
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 6
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 6
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 5
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 5
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 6
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 6
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 6
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 7
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 7
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 6
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 6
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 7
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 6
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 5
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 5
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 5
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 5
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 6
[1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 7
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 6
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 8
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 8
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 7
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 7
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 7
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 6
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 6
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 7
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 8
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 9
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 7
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 8
[] => [] => [] => [1] => 0
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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
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
Map
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.