Identifier
Values
[1] => [1,0] => [1,0] => [2,1] => 1
[2] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 1
[1,1] => [1,1,0,0] => [1,0,1,0] => [3,1,2] => 1
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 1
[2,2] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 1
[5] => [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] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 1
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 1
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 1
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 1
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 1
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 1
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => 1
[] => [] => [] => [1] => 1
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of cycles in the cycle decomposition of a permutation.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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.